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• ill THEOBY OF FUNCTIONS OF A COMPLEX VARIABLE. Itonlion: C. J. CLAY AND SONS, CAMBEIDGE UNIVEESITY PEESS WAEEHOUSE, AVE MAEIA LANE. CAMBEIDGE : DEIGHTON, BELL, AKD CO. LEIPZIG : F. A. BROCKHAUS. NEW YORK: MACMILLAN AND CO. THEOEY OF FUNCTIONS OF A COMPLEX VARIABLE BY A. R FORSYTE, So.D., F.RS., FELLOW OF TRINITY COLLEGE, CAMBRIDGE. CAMBEIDGE: AT THE UNIVERSITY PRESS. 1893 All rights reserved. . Mtth. U. 01. PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS. PEEFACE. AMONG the many advances in the progress of mathematical XlL science during the last forty years, not the least remarkable are those in the theory of functions. The contributions that are still being made to it testify to its vitality : all the evidence points to the continuance of its growth. And, indeed, this need cause no surprise. Few subjects can boast such varied processes, based upon methods so distinct from one another as are those originated by Cauchy, by Weierstrass, and by Biemann. Each of these methods is sufficient in itself to provide a complete development ; combined, they exhibit an unusual wealth of ideas and furnish unsurpassed resources in attacking new problems. It is difficult to keep pace with the rapid growth of the literature which is due to the activity of mathematicians, especially of continental mathematicians : and there is, in con sequence, sufficient reason for considering that some marshalling of the main results is at least desirable and is, perhaps, necessary. Not that there is any dearth of treatises in French and in German : but, for the most part, they either expound the pro cesses based upon some single method or they deal with the discussion of some particular branch of the theory. 814033 PREFACE The present treatise is an attempt to give a consecutive account of what may fairly be deemed the principal branches of the whole subject. It may be that the next few years will see additions as important as those of the last few years : this account would then be insufficient for its purpose, notwithstanding the breadth of range over which it may seem at present to extend. My hope is that the book, so far as it goes, may assist mathe maticians, by lessening the labour of acquiring a proper knowledge of the subject, and by indicating the main lines, on which recent progress has been achieved. No apology is offered for the size of the book. Indeed, if there were to be an apology, it would rather be on the ground of the too brief treatment of some portions and the omissions of others. The detail in the exposition of the elements of several important branches has prevented a completeness of treatment of those branches : but this fulness of initial explanations is deliberate, my opinion being that students will thereby become better qualified to read the great classical memoirs, by the study of which effective progress can best be made. And limitations of space have compelled me to exclude some branches which other wise would have found a place. Thus the theory of functions of a real variable is left undiscussed : happily, the treatises of Dini, Stolz, Tannery and Chrystal are sufficient to supply the omission. Again, the theory of functions of more than one complex variable receives only a passing mention ; but in this case, as in most cases, where the consideration is brief, references are given which will enable the student to follow the development to such extent as he may desire. Limitation in one other direction has been imposed : the treatise aims at dealing with the general theory of functions and it does not profess to deal with special classes of functions. I have not hesitated to use examples of special classes : but they are used merely as illustrations of the general theory, and references are given to other treatises for the detailed exposition of their properties. PREFACE Vll The general method which is adopted is not limited so that it may conform to any single one of the three principal inde pendent methods, due to Cauchy, to Weierstrass and to Biemann respectively : where it has been convenient to do so, I have combined ideas and processes derived from different methods. The book may be considered as composed of five parts. The first part, consisting of Chapters I — VII, contains the theory of uniform functions : the discussion is based upon power- series, initially connected with Cauchy's theorems in integration, and the properties established are chiefly those which are con tained in the memoirs of Weierstrass and Mittag-Leffler. The second part, consisting of Chapters VIII — XIII, contains the theory of multiform functions, and of uniform periodic functions which are derived through the inversion of integrals of algebraic functions. The method adopted in this part is Cauchy's, as used by Briot and Bouquet in their three memoirs and in their treatise on elliptic functions : it is the method that has been followed by Hermite and others to obtain the properties of various kinds of periodic functions. A chapter has been devoted to the proof of Weierstrass's results relating to functions that possess an addition-theorem. The third part, consisting of Chapters XIV — XVIII, contains the development of the theory of functions according to the method initiated by Biemann in his memoirs. The proof which is given of the existence-theorem is substantially due to Schwarz ; in the rest of this part of the book, I have derived great assist ance from Neumann's treatise on Abelian functions, from Fricke's treatise on Klein's theory of modular functions, and from many memoirs by Klein. The fourth part, consisting of Chapters XIX and XX, treats of conformal representation. The fundamental theorem, as to the possibility of the conformal representation of surfaces upon one another, is derived from the existence-theorem : it is a curious fact that the actual solution, which has been proved to exist in general, F. b Vlll PREFACE has been obtained only for cases in which there is distinct limitation. The fifth part, consisting of Chapters XXI and XXII, contains an introduction to the theory of Fuchsian or automorphic functions, based upon the researches of Poincare and Klein : the discussion is restricted to the elements of this newly-developed theory. The arrangement of the subject-matter, as indicated in this abstract of the contents, has been adopted as being the most convenient for the continuous exposition of the theory. But the arrangement does not provide an order best adapted to one who is reading the subject for the first time. I have therefore ventured to prefix to the Table of Contents a selection of Chapters that will probably form a more suitable introduction to the subject for such a reader ; the remaining Chapters can then be taken in an order determined by the branch of the subject which he wishes to follow out. In the course of the preparation of this book, I have consulted many treatises and memoirs. References to them, both general and particular, are freely made : without making precise reserva tions as to independent contributions of my own, I wish in this place to make a comprehensive acknowledgement of my obligations to such works. A number of examples occur in the book : most of them are extracted from memoirs, which do not lie close to the direct line of development of the general theory but contain results that provide interesting special illustrations. My inten tion has been to give the author's name in every case where a result has been extracted from a memoir : any omission to do so is due to inadvertence. Substantial as has been the aid provided by the treatises and memoirs to which reference has just been made, the completion of the book in the correction of the proof-sheets has been rendered easier to me by the unstinted and untiring help rendered by two friends. To Mr William Burnside, M.A., formerly Fellow of PREFACE Pembroke College, Cambridge, and now Professor of Mathematics at the Royal Naval College, Greenwich, I am under a deep debt of gratitude : he has used his great knowledge of the subject in the most generous manner, making suggestions and criticisms that have enabled me to correct errors and to improve the book in many respects. Mr H. M. Taylor, M. A., Fellow of Trinity College, Cambridge, has read the proofs with great care : the kind assist ance that he has given me in this way has proved of substantial service and usefulness in correcting the sheets. I desire to recognise most gratefully my sense of the value of the work which these gentlemen have done. It is but just on my part to state that the willing and active co-operation of the Staff of the University Press during the pro gress of printing has done much to lighten my labour. It is, perhaps, too ambitious to hope that, on ground which is relatively new to English mathematics, there will be freedom from error or obscurity and that the mode of presentation in this treatise will command general approbation. In any case, my aim has been to produce a book that will assist mathematicians in acquiring a knowledge of the theory of functions : in proportion as it may prove of real service to them, will be my reward. A. R. FORSYTE. TRINITY COLLEGE, CAMBRIDGE. 25 February, 1893. CONTENTS. The following course is recommended, in the order specified, to those who are reading the subject for the first time : The theory of uniform functions, Chapters I— V ; Conformal representation, Chapter XIX ; Multiform functions and uniform periodic functions, Chapters VIII— XI ; Riemanris surfaces, and Riemann's theory of algebraic functions and their integrals, Chapters XIV— XVI, XVIII. CHAPTER I. GENERAL INTRODUCTION. §§ PAGE 1—3. The complex variable and the representation of its variation by points in a plane , 4. Neumann's representation by points on a sphere ... 4 5. Properties of functions assumed known ... Q 6, 7. The idea of complex functionality adopted, with the conditions neces sary and sufficient to ensure functional dependence ... 6 8. Riemann's definition of functionality ... g 9. A functional relation between two complex variables establishes the geometrical property of conformal representation of their planes . 10 10, 11. Relations between the real and the imaginary parts of a function of z 11 12, 13. Definitions and illustrations of the terms monogenic, uniform, multiform, branch, branch-point, holomorphic, zero, pole, meromorphic . . . 14 CHAPTER II. INTEGRATION OF UNIFORM FUNCTIONS. 14, 15. Definition of an integral with complex variables ; inferences . . . ' 18 16. Proof of the lemma I I (^ - £ \ dxdy=\(pdx -\-qdy), under assigned I \ fll'1 (it I I J •* J. •/ f ' O conditions 21 CONTENTS §§ PAGE 17, 18. The integral \f(z)dz round any simple curve is zero, when f(z) is Cz holomorphic within the curve; and I /(*)<& is a holomorphic J a function when the path of integration lies within the curve . . 23 19. The path of integration of a holomorphic function can be deformed without changing the value of the integral ..... 26 20—22. The integral =— . I '— '- dz, round a curve enclosing a, is /(a) when 27rt J z — a f(z) is a holomorphic function within the curve; and the integral J_ [ /(*) dz is — ,—^. Superior limit for the modulus of 27rt J(z-a)n + 1 n\ dan the nth derivative of /(a) in terms of the modulus of /(a) . . 27 23. The path of integration of a meromorphic function cannot be deformed across a pole without changing the value of the integral. . . 34 24. The integral of any function (i) round a very small circle, (ii) round a very large circle, (iii) round a circle which encloses all its infinities and all its branch-points ......... 35 25. Special examples ............ •• CHAPTER III. EXPANSIONS OF FUNCTIONS IN SERIES OF POWERS. 26, 27. Cauchy's expansion of a function in positive powers of z - a ; with re marks and inferences 43 28—30. Laurent's expansion of a function in positive and negative powers of z - a ; with corollary 47 31. Application of Cauchy's expansion to the derivatives of a function . 51 32, 33. Definition of an ordinary point of a function, of the domain of an ordinary point, of an accidental singularity, and of an essential singularity .......••••• 52 34, 35. Continuation of a function by means of elements over its region of continuity 54 36. Schwarz's theorem on symmetric continuation across the axis of real quantities 57 CHAPTER IV. UNIFORM FUNCTIONS, PARTICULARLY THOSE WITHOUT ESSENTIAL SINGULARITIES. 37. A function, constant over a continuous series of points, is constant everywhere in its region of continuity 59 38, 39. The multiplicity of a zero, which is an ordinary point, is finite; and a multiple zero of a function is a zero of its first derivative . . 61 CONTENTS Xlll §§ PAGE 40. A function, that is not a constant, must have infinite values . . 63 41, 42. Form of a function near an accidental singularity 64 43, 44. Poles of a function are poles of its derivatives ..... 66 45, 46. A function, which has infinity for its only pole and has no essential singularity, is an algebraical polynomial ...... 69 47. Integral algebraical and integral transcendental functions ... 70 48. A function, all the singularities of which are accidental, is an algebraical meromorphic function .......... 71 CHAPTER V. TRANSCENDENTAL INTEGRAL UNIFORM FUNCTIONS. 49, 50. Construction of a transcendental integral function with assigned zeros a1? a2, a3, ..., when an integer s can be found such that 2|an|~8 is a converging series 74 51. Weierstrass's construction of a function with any assigned zeros . . 77 52, 53. The most general form of function with assigned zeros and having its single essential singularity at 0=00 . . . . . . 80 54. Functions with the singly-infinite system of zeros given by ;Z = TO<B, for integral values of m 82 55 — 57. Weierstrass's o--function with the doubly-infinite system of zeros given by z=ma> + m'a>, for integral values of TO and of TO' . . . . 84 58. A function cannot exist with a triply-infinite arithmetical system of zeros 88 59, 60. Class (genre) of a function 89 61. Laguerre's criterion of the class of a function 91 CHAPTER VI. FUNCTIONS WITH A LIMITED NUMBER OF ESSENTIAL SINGULARITIES. 62. Indefiniteness of value of a function at an essential singularity . . 94 63. A function is of the form O { — =- ) + P (z — 6) in the vicinity of an essen- \»— o/ tial singularity at b, a point in the finite part of the plane . . 96 64, 65. Expression of a function with n essential singularities as a sum of n functions, each with only one essential singularity .... 99 66, 67. Product-expression of a function with n essential singularities and no zeros or accidental singularities 101 68 — 71. Product-expression of a function with n essential singularities and with assigned zeros and assigned accidental singularities ; with a note on the region of continuity of such a function . . . .104 xiv CONTENTS CHAPTER VII. FUNCTIONS WITH UNLIMITED ESSENTIAL SINGULARITIES, AND EXPANSION IN SERIES OF FUNCTIONS. §§ 1>AGE 72. Mittag-Leffler's theorem on functions with unlimited essential singu larities, distributed over the whole plane 112 73. Construction of subsidiary functions, to be terms of an infinite sum . 113 74_76. Weierstrass's proof of Mittag-Leffler's theorem, with the generalisation of the form of the theorem 114 77, 78. Mittag-Leffler's theorem on functions with unlimited essential singu larities, distributed over a finite circle 117 79. Expression of a given function in Mittag-Leffler's form .... 123 80. General remarks on infinite series, whether of powers or of functions . 126 81. A series of powers, in a region of continuity, represents one and only one function ; it cannot be continued beyond a natural limit . . 128 82. Also a series of functions : but its region of continuity may consist of distinct parts 129 83. A series of functions does not necessarily possess a derivative at points on the boundary of any one of the distinct portions of its region of continuity ........••• 133 84. A series of functions may represent different functions in distinct parts of its region of continuity ; Tannery's series 136 85. Construction of a function which represents different assigned functions in distinct assigned parts of the plane . . . . . .138 86. Functions with a line of essential singularity 139 87. Functions with an area of essential singularity or lacunary spaces . 141 88. Arrangement of singularities of functions into classes and species . . 146 CHAPTER VIII. MULTIFORM FUNCTIONS. 89. Branch-points and branches of functions 149 90. Branches obtained by continuation: path of variation of independent variable between two points can be deformed without affecting a branch of a function if it be not made to cross a branch-point . 150 91. If the path be deformed across a branch-point which affects the branch, then the branch is changed I55 92. The interchange of branches for circuits- round a branch-point is cyclical 156 93. Analytical form of a function near a branch-point 157 94. Branch-points of a function defined by an algebraical equation in their relation to the branches : definition of algebraic function . . 161 95, 96. Infinities of an algebraic function 163 CONTENTS xv PAGE 97. Determination of the branch-points of an algebraic function, and of the cyclical systems of the branches of the function ... 168 98. Special case, when the branch-points are simple : their number . . 174 99. A function, with n branches and a limited number of branch-points and singularities, is a root of an algebraical equation of degree n. . 175 CHAPTER IX. PERIODS OF DEFINITE INTEGRALS, AND PERIODIC FUNCTIONS IN GENERAL. 100. Conditions under which the path of variation of the integral of a multiform function can be deformed without changing the value of the integral ....... J§Q 101. Integral of a multiform function round a small curve enclosing a branch-point ....... 183 102. Indefinite integrals of uniform functions with accidental singularities ; fdz f dz j i ' 2 ..... • ..... 184 103. Hermite's method of obtaining the multiplicity in value of an integral; sections in the plane, made to avoid the multiplicity . . .185 104. Examples of indefinite integrals of multiform functions ; \wdz round any loop, the general value of J(l - 22) ~ * dz, of J{1 - z2) (1 - k^}} ~ * dz, and of 5{(z-el)(z-e2)(g-es)}-*dz ....... 189 105. Graphical representation of simply-periodic and of doubly-periodic functions ....... 198 106. The ratio of the periods of a uniform doubly-periodic function is not real ............. 200 107, 108. Triply-periodic uniform functions of a single variable do not exist . 202 109. Construction of a fundamental parallelogram for a uniform doubly- periodic function ....... 205 110. An integral, with more periods than two, can be made to assume any value by a modification of the path of integration between the limits ........ 208 CHAPTER X. SIMPLY-PERIODIC AND DOUBLY-PERIODIC FUNCTIONS. 2rrzi 111. Simply-periodic functions, and the transformation Z=e w . ;' • . 211 112. Fourier's series and simply-periodic functions 213 113, 114. Properties of simply-periodic functions without essential singularities in the finite part of the plane 214 115. Uniform doubly-periodic functions, without essential singularities in the finite part of the plane 218 116. Properties of uniform doubly-periodic functions 219 CONTENTS §§ 117. The zeros and the singularities of the derivative of a doubly-periodic function of the second order . . 231 118, 119. Kelations between homoperiodic functions . . ... . • 233 CHAPTER XL DOUBLY-PERIODIC FUNCTIONS OF THE SECOND ORDER. 120 121. Formation of an uneven function with two distinct irreducible in finities; its addition-theorem 243 122, 123. Properties of Weierstrass's o-function . . . . • • 247 124. Introduction of f (2) and of Q(z) 250 125, 126. Periodicity of the function #> (z), with a single irreducible infinity of degree two; the differential equation satisfied by the function #> (2) 251 127. Pseudo-periodicity of f(«) • 255 128. Construction of a doubly-periodic function in terms of f (z) and its derivatives • . - . • • 256 129. On the relation qw'- j/eo = ±%iri 25>7 130. Pseudo-periodicity of a (z) • 259 131. Construction of a doubly-periodic function as a product of o-functions ; with examples 259 132. On derivatives of periodic functions with regard to the invariants #2 and £3 * ' ' lfK 133 135. Formation of an even function of either class 266 CHAPTER XII. PSEUDO-PERIODIC FUNCTIONS. 136. Three kinds of pseudo-periodic functions, with the characteristic equa tions 273 137, 138. Hermite's and Mittag-Leffler's expressions for doubly-periodic functions of the second kind 275 139. The zeros and the infinities of a secondary function . . - . . 280 140, 141. Solution of Lamp's differential equation 281 142. The zeros and the infinities of a tertiary function .... 286 143. Product-expression for a tertiary function 287 144—146. Two classes of tertiary functions ; Appell's expressions for a function of each class as a sum of elements 288 147. Expansion in trigonometrical series 293 148. Examples of other classes of pseudo-periodic functions . . . 295 CONTENTS Xvii CHAPTER XIII. FUNCTIONS POSSESSING AN ALGEBRAICAL ADDITION-THEOREM. §§ PAGE 149. Definition of an algebraical addition-theorem 297 150. A function defined by an algebraical equation, the coefficients of which are algebraical functions, or simply-periodic functions, or doubly-periodic functions, has an algebraical addition-theorem . 297 151 — 154. A function possessing an algebraical addition-theorem is either algebraical, simply-periodic or doubly-periodic, having in each instance only a finite number of values for an argument . . 300 155, 156. A function with an algebraical addition-theorem can be defined by a differential equation of the first order, into which the independent variable does not explicitly enter 309 CHAPTER XIV. CONNECTIVITY OF SURFACES. 157—159. Definitions of connection, simple connection, multiple connection, cross cut, loop-cut . . . .'.•-., 312 160. Relations between cross-cuts and connectivity 315 161. Relations between loop-cuts and connectivity 320 162. Effect of a slit .321 163, 164. Relations between boundaries and connectivity 322 165. Lhuilier's theorem on the division of a connected surface into curvilinear polygons 325 166. Definitions of circuit, reducible, irreducible, simple, multiple, compound, reconcileable 327 167, 168. Properties of a complete system of irreducible simple circuits on a surface, in its relation to the connectivity 328 169. Deformation of surfaces 332 170. Conditions of equivalence for representation of the variable . . 333 CHAPTER XV. RIEMANN'S SURFACES. 171. Character and general description of a Riemann's surface . - .. •. 336 172. Riemann's surface associated with an algebraical equation . . .338 173. Sheets of the surface are connected along lines, called branch-lines . 338 174. Properties of branch-lines 340 175, 176. Formation of system of branch-lines for a surface ; with examples . 341 177. Spherical form of Riemann's surface . . . 34(5 XV111 CONTENTS §§ PAGE 178. The connectivity of a Eiemann's surface 347 179. Irreducible circuits : examples of resolution of Riemann's surfaces into surfaces that are simply connected 350 180, 181. General resolution of a Riemann's surface 353 182. A Riemann's %-sheeted surface when all the branch-points are simple 355 183, 184. On loops, and their deformation 356 185. Simple cycles of Clebsch and Gordan 359 186 — 189. Canonical form of Riemann's surface when all the branch -points are simple, deduced from theorems of Luroth and Clebsch. . . 361 190. Deformation of the surface 365 191. Remark on uniform algebraical transformations 367 CHAPTER XVI. ALGEBRAIC FUNCTIONS AND THEIR INTEGRALS. 192. Two subjects of investigation 368 193, 194. Determination of the most general uniform function of position on a Riemann's surface .......... 369 195. Preliminary lemmas in integration on a Riemann's surface . . . 372 196, 197. Moduli of periodicity for cross-cuts in the resolved surface . . . 373 198. The number of linearly independent moduli of periodicity is equal to the number of cross-cuts, which are necessary for the resolution of the surface into one that is simply connected .... 378 199. Periodic functions on a Riemann's surface, with examples . . . 379 200. Integral of the most general uniform function of position on a Riemann's surface . . . ; 387 201. Integrals, everywhere finite on the surface, connected with the equa tion w*=S(z) 388 202 — 204. Infinities of integrals on the surface connected with the algebraical equation f (w, z) = 0, when the equation is geometrically interpret- able as the equation of a (generalised) curve of the nth order . 388 205, 206. Integrals of the first kind connected with/(w, z) = 0, Demg functions that are everywhere finite : the number of such integrals, linearly independent of one another : they are multiform functions . . 394 207, 208. Integrals of the second kind connected with f (w, z) = 0, being func tions that have only algebraical infinities; elementary integral of the second kind .......... 396 209. Integrals of the third kind connected with/(w, z) = 0, being functions that have logarithmic infinities 400 210, 211. An integral of the third kind cannot have less than two logarithmic infinities ; elementary integral of the third kind .... 401 CONTENTS CHAPTER XVII. SCHWARZ'S PROOF OF THE EXISTENCE-THEOREM. §§ PAGE 212, 213. Existence of functions on a Riemann's surface; initial limitation of the problem to the real parts u of the functions . ... . . 405 214. Conditions to which u, the potential function, is subject . . . 407 215. Methods of proof : summary of Schwarz's investigation . . . 408 216 — 220. The potential-function u is uniquely determined for a circle by the gene ral conditions and by the assignment of finite boundary values . 410 221. Also for any plane area, on which the area of a circle can be con- formally represented 423 222. Also for any plane area which can be obtained by a topological com bination of areas, having a common part and each conformally representable on the area of a circle 425 223. Also for any area on a Riemann's surface in which a branch-point occurs ; and for any simply connected surface .... 428 224 — 227. Real functions exist on a Riemann's surface, everywhere finite, and having arbitrarily assigned real moduli of periodicity . . . 430 228. And the number of the linearly independent real functions thus ob tained is 2p ........... 434 229. Real functions exist with assigned infinities on the surface and assigned real moduli of periodicity. Classes of functions of the complex variable proved to exist on the Riemann's surface . . 435 CHAPTER XVIII. APPLICATIONS OF THE EXISTENCE-THEOREM. 230. Three special kinds of functions on a Riemann's surface . . . 437 231 — 233. Relations between moduli of functions of the first kind and those of functions of the second kind 439 234. The number of linearly independent functions of the first kind on a Riemann's surface of connectivity 2/; + l is p 235. Normal functions of the first kind ; properties of their moduli . 236. Normal elementary functions of the second kind : their moduli . 237, 238. Normal elementary functions of the third kind : their moduli : inter change of arguments and parametric points 449 239. The inversion-problem for functions of the first' kind .... 453 240. Algebraical functions on a Riemann's surface without infinities at the branch-points but only at isolated ordinary points on the surface : Riemann-Roch's theorem : the smallest number of singularities that such functions may possess . . . . . . .457 241. A class of algebraic functions infinite only at branch-points . . 460 242. Fundamental equation associated with an assigned Riemann's surface 462 XX CONTENTS §§ PAGE 243. Appell's factorial functions on a Riemann's surface : their multipliers at the cross-cuts 464 244, 245. Expression for a factorial function with assigned zeros and assigned infinities; relations between zeros and infinities of a factorial function . . ... . . .' . • • • 466 246. Functions defined by differential equations of the form / ( w, -y- ) = 0 470 \ **&) 247 — 249. Conditions that the function should be a uniform function of z. . 471 250, 251. Classes of uniform functions that can be so defined, with criteria of discrimination . . • , • • • • • • • • 476 (dw\s ~T~ ) =/ (w) .... 482 az j CHAPTER XIX. CONFORMAL REPRESENTATION : INTRODUCTORY. 253. A relation between complex variables is the most general relation that secures conformal similarity between two surfaces .... 491 254. One of the surfaces for conformal representation may, without loss of generality, be taken to be a plane 495 255, 256. Application to surfaces of revolution ; in particular, to a sphere, so as to obtain maps .......... 496 257. Some examples of conformal representation of plane areas, in par ticular, of areas that can be conformally represented on the area of a circle 501 258. Linear homographic transformations (or substitutions) of the form w = ,: their fundamental properties 512 cz + d 259. Parabolic, elliptic, hyperbolic and loxodromic substitutions . . . 517 260. An elliptic substitution is either periodic or infinitesimal : substitu tions of the other classes are neither periodic nor infinitesimal . 521 261. A linear substitution can be regarded geometrically as the result of an even number of successive inversions of a point with regard to circles . ......... 523 CHAPTER XX. CONFORMAL REPRESENTATION : GENERAL THEORY. 262. Riemann's theorem on the conformal representation of a given area upon the area of a circle with unique correspondence . . . 525 263, 264. Proof of Riemann's theorem : how far the functional equation is algebraically determinate 526 265, 266. The method of Beltrami and Cayley for the construction of the functional equation for an analytical curve 530 CONTENTS XXI §§ PAGE 267, 268. Conformal representation of a convex rectilinear polygon upon the half-plane of the variable 537 269. The triangle, and the quadrilateral, conformally represented . . 543 270. A convex curve, as a limiting case of a polygon .... 548 271, 272. Conformal representation of a convex figure, bounded by circular arcs : the functional relation is connected with a linear differential equation of the second order ........ 549 273. Conformal representation of a crescent ....... 554 274 — 276. Conformal representation of a triangle, bounded by circular arcs . 555 277 — 279. Relation between the triangle, bounded by circles, and the stereographic projection of regular solids inscribed in a sphere .... 563 280. On families of plane algebraical curves, determined as potential-curves by a single parameter u + vi : the forms of functional relation ), which give rise to such curves .... 575 CHAPTER XXI. GROUPS OF LINEAR SUBSTITUTIONS. 281. The algebra of group-symbols 582 282. Groups, which are considered, are discontinuous and have a finite number of fundamental substitutions 584 283, 284. Anharmonic group : group for the modular-functions, and division of the plane of the variable to represent the group .... 586 285, 286. Fuchsian groups : division of plane into convex curvilinear polygons : polygon of reference 591 287. Cycles of angular points in a curvilinear polygon .... 595 288, 289. Character of the division of the plane : example .... 599 290. Fuchsian groups which conserve a fundamental circle . . . 602 291. Essential singularities of a group, and of the automorphic functions determined by the group ........ 605 292, 293. Families of groups : and their class 606 294. Kleinian groups : the generalised equations connecting two points in space "... . . .609 295. Division of plane and division of space, in connection with Kleinian groups 613 296. Example of improperly discontinuous group 615 CHAPTER XXII. AUTOMORPHIC FUNCTIONS. 297. Definition of automorphic functions . . . . . '- . .619 298. Examples of functions, automorphic for finite discrete groups of sub stitutions 620 299. Cayley's analytical relation between stereographic projections of posi tions of a point on a rotated sphere 620 XX11 CONTENTS §§ PAGE 300. Polyhedral groups ; in particular, the dihedral group, and the tetra- hedral group 623 301, 302. The tetrahedral functions, and the dihedral functions . . . 628 303. Special illustrations of infinite discrete groups, from the elliptic modular-functions 633 304. Division of the plane, and properties of the fundamental polygon of reference, for any infinite discrete group that conserves a funda mental circle ........... 637 305, 306. Construction of Thetafuchsian functions, pseudo-automorphic for an infinite group of substitutions 641 307. Relations between the number of irreducible zeros and the number of irreducible poles of a pseudo-automorphic function, constructed with a rational algebraical meromorphic function as element . 645 308. Construction of automorphic functions ....... 650 309. The number of irreducible points, for which an automorphic function acquires an assigned value, is independent of the value . . 651 310. Algebraical relations between functions, automorphic for a group : application of Riemanu's theory of functions .... 653 311. Connection between automorphic functions and linear differential equations ; with illustrations from elliptic modular-functions . 654 GLOSSARY OF TECHNICAL TERMS . . . . ; . . ' , . 659 INDEX OF AUTHORS QUOTED . . . . - . . . . . 662 GENERAL INDEX 664 CHAPTER I. GENERAL INTRODUCTION. 1. ALGEBRAICAL operations are either direct or inverse. Without entering into a general discussion of the nature of irrational and of imaginary quantities, it will be sufficient to point out that direct algebraical operations on numbers that are positive and integral lead to numbers of the same character; and that inverse algebraical operations on numbers that are positive and integral lead to numbers, which may be negative or fractional or irrational, or to numbers which may not even fall within the class of real quantities. The simplest case of occurrence of a quantity, which is not real, is that which arises when the square root of a negative quantity is required. Combinations of the various kinds of quantities that may occur are of the form x + iy, where x and y are real and i, the non-real element of the quantity, denotes the square root of - 1. It is found that, when quantities of this character are subjected to algebraical operations, they always lead to quantities of the same formal character; and it is therefore inferred that the most general form of algebraical quantity is x + iy. Such a quantity ic + iy, for brevity denoted by z, is usually called a complex variable*; it therefore appears that the complex variable is the most general form of algebraical quantity which obeys the fundamental laws of ordinary algebra. 2. The most general complex variable is that, in which the constituents x and y are independent of one another and (being real quantities) are separately capable of assuming all values from - oo to + oo ; thus a doubly- infinite variation is possible for the variable. In the case of a real variable, it is convenient to use the customary geometrical representation by measure ment of distance along a straight line; so also in the case of a complex * The conjugate complex, viz. x - iy, is frequently denoted by za. F. 2 GEOMETRICAL REPRESENTATION OF [2. variable, it is convenient to associate a geometrical representation with the algebraical expression ; and this is the well-known representation of the variable ac + iy by means of a point with coordinates x and y referred to rectangular axes*. The complete variation of the complex variable z is represented by the aggregate of all possible positions of the associated point, which is often called the point z ; the special case of real variables being evidently included in it because, when y = 0, the aggregate of possible points is the line which is the range of geometrical variation of the real "variable. • The variation of z is said to be continuous when the variations of x and y are contiguous. Continuous variation of z between two given values will thus be represented by continuous variation in the position of the point z, that is, by a continuous curve (not necessarily of continuous curvature) between the points corresponding to the two values. But since an infinite number of curves can be drawn between two points in a plane, continuity of line is not sufficient to specify the variation of the complex variable ; and, in order to indicate any special mode of variation, it is necessary to assign, either explicitly or implicitly, some determinate law connecting the variations of x and y or, what is the same thing, some determinate law connecting x and y. The analytical expression of this law is the equation of the curve which represents the aggregate of values assumed by the variable between the two given values. In such a case the variable is often said to describe the part of the curve between the two points. In particular, if the variable resume its initial value, the representative point must return to its initial position ; and then the variable is said to describe the whole curve -f-. When a given closed curve is continuously described by the variable, there are two directions in which the description can take place. From the analogy of the description of a straight line by a point representing a real variable, one of these directions is considered as positive and the other * This method of geometrical representation of imaginary quantities, ordinarily assigned to Gauss, was originally developed by Argand who, in 1806, published his " Essai sur une maniere de representer les quantites imaginaires dans les constructions geometriques." This tract was republished in 1874 as a second edition (Gauthier-Villars) ; an interesting preface is added to it by Hoiiel, who gives an account of the earlier history of the publications associated with the theory. Other references to the historical development are given in Chrystal's Text-book of Algebra, vol. i, pp. 248, 249; in Holzmiiller's Einfilhrung in die Theorie der isogonalen Venvandschaften und dcr conformen Abbildungen, verbunden mit Anwendungen auf mathematische Physik, pp. 1 — 10, 21 — 23 ; in Schlomilch's Compendium der hoheren Analysis, vol. ii, p. 38 (note) ; and in Casorati, Teorica delle funzioni di variabili complesse, only one volume of which was published. In this connection, an article by Cayley (Quart. Journ. of Math,, vol. xxii, pp. 270 — 308) may be consulted with advantage. t In these elementary explanations, it is unnecessary to enter into any discussion of the effects caused by the occurrence of singularities in the curve. 2-] THE COMPLEX VARIABLE Fig. 1. as negative. The usual convention under which one of the directions is selected as the positive direction depends upon the conception that the curve is the boundary, partial or complete, of some area ; under it, that direction is taken to be positive which is such that the bounded area lies to the left of the direction of description. It is easy to see that the same direction is taken to be positive under an equivalent convention which makes it related to the normal drawn outwards from the bounded area in the same way as the positive direction of the axis of y is to the positive direction of the axis of x in plane coordinate geometry. Thus in the figure (fig. 1), the positive direction of description of the outer curve for the area included by it is DEF; the positive direction of description of the inner curve for the area without it (say, the area excluded by it) is AGB ; and for the area between the curves the positive direction of description of the boundary, which consists of two parts, is DEF, ACB. 3. Since the position of a point in a plane can be determined by means of polar coordinates, it is convenient in the discussion of complex variables to introduce two quantities corresponding to polar coordinates. In the case of the variable z, one of these quantities is (#2 + yn-)l, the positive sign being always associated with it ; it is called the modulus* of the variable and it is denoted, sometimes by mod. z, sometimes by \z . The other is 0, the angular coordinate of the point z ; it is called the argument (and, less frequently, the amplitude) of the variable. It is measured in the trigonometrically positive sense, and is determined by the equations <K=\Z\ cos 6, y= z\ sin#, so that z= z\eei. The actual value depends upon the way in which the variable has acquired its value ; when variation of the argument is considered, its initial value is usually taken to lie between 0 and 2?r or, less frequently, between -TT and +TT. As z varies in position, the values of \z\ and 6 vary. When z has completed a positive description of a closed curve, the modulus of z returns to the initial value whether the origin Fig. 2. Der absolute Metro,;) is often used by German writers. 1—2 GREAT VALUES OF [3. be without, within or on the curve. The argument of z resumes its initial value, if the origin 0' (fig. 2) be without the curve ; but, if the origin 0 be within the curve, the value of the argument is increased by 2-rr when z returns to its initial position. If the origin be on the curve, the argument of z undergoes an abrupt change by TT as z passes through the origin ; and the change is an increase or a decrease according as the variable approaches its limiting position on the curve from without or from within. No choice need be made between these alternatives; for care is always exercised to choose curves which do not introduce this element of doubt. 4. Representation on a plane is obviously more effective for points at a finite distance from the origin than for points at a very great distance. One method of meeting the difficulty of representing great values is to introduce a new variable z1 given by z'z=\\ the part of the new plane for z which lies quite near the origin corresponds to the part of the old plane for z which is very distant. The two planes combined give a complete representation of variation of the complex variable. Another method, in many ways more advantageous, is as follows. Draw a sphere of unit diameter, touching the 2-plane at the origin 0 (fig. 3) on the under side: join a point z in the plane to 0', the other extremity of the diameter through 0, by a straight line cutting the sphere in Z. Then Z is a unique representative of z, that is, a single point on the sphere corresponds to a single point on the plane : and therefore the variable can be represented on the surface of the sphere. With this mode of Fig. 3. representation, 0' evidently corresponds to an infinite value of z : and points at a very great distance in the 2-plane are represented by points in the immediate vicinity of 0' on the sphere. The sphei-e thus has the advantage of putting in evidence a part of the surface jn which the variations of 4.] THE COMPLEX VARIABLE 5 great values of z can be traced*, and of exhibiting the uniqueness of z — oo as a value of the variable, a fact that is obscured in the represen tation on a plane. The former method of representation can be deduced by means of the sphere. At 0' draw a plane touching the sphere : and let the straight line OZ cut this plane in z'. Then z is a point uniquely determined by Z and therefore uniquely determined by z. In this new /-plane take axes parallel to the axes in the 2-plane. The points z and / move in the same direction in space round 00' as an axis. If we make the upper side of the 2-plane correspond to the lower side of the /-plane, and take the usual positive directions in the planes, being the positive trigonometrical directions for a spectator looking at the surface of the plane in which the description takes place, we have these directions indicated by the arrows at 0 and at 0' respectively, so that the senses of positive rotations in the two planes are opposite in space. Now it is evident from the geometry that Oz and O'z' are parallel ; hence, if 0 be the argument of the point z and & that of the point z so that 6 is the angle from Ox to Oz and 6' the angle from O'x' to O'z, we have 6 + ff = ZTT. Oz 00' Further, by similar triangles, -^-t = ^-f , that is, Oz . O'z' = OO'2 = 1. Now, if z and z' be the variables, we have z=0z.eei, z'=0'z'.effi, so that zz'=0z.0'z' .e^s'^ = 1, which is the former relation. The /-plane can therefore be taken as the lower side of a plane touching the sphere at 0' when the 2-plane is the upper side of a plane touching it at 0. The part of the 2-plane at a very great distance is represented on the sphere by the part in the immediate vicinity of 0' : and this part of the sphere is represented on the /-plane by its portion in the immediate vicinity of 0', which therefore is a space wherein the variations of infinitely great values of z can be traced. But it need hardly be pointed out that any special method of represent ation of the variable is not essential to the development of the theory of functions ; and, in particular, the foregoing representation of the variable, when it has very great values, merely provides a convenient method of dealing with quantities that tend to become infinite in magnitude. * This sphere is sometimes called Neumann's sphere; it is used by him for the representation of the complex variable throughout his treatise Vorlesungen uber Riemann'a Theorie der AlcVschen Integrate (Leipzig, Teubner, '2nd edition, 1884). 6 CONDITIONS OF [5. 5. The simplest propositions relating to complex variables will be assumed known. Among these are, the geometrical interpretation of opera tions such as addition, multiplication, root-extraction ; some of the relations of complex variables occurring as roots of algebraical equations with real coefficients; the elementary properties of functions of complex variables which are algebraical and integral, or exponential, or circular functions; and simple tests of convergence of infinite series and of infinite products*. 6. All ordinary operations effected on a complex variable lead, as already remarked, to other complex variables; and any definite quantity, thus obtained by operations on z, is necessarily a function of z. But if a complex variable w be given as a complex function of x and y without any indication of its source, the question as to whether w is or is not a function of z requires a consideration of the general idea of functionality. It is convenient to postulate u + iv as a form of the complex variable w, where u and v are real. Since w is initially unrestricted in variation, we may so far regard the quantities u and v as independent and therefore as any functions of x and y, the elements involved in z. But more explicit expressions for these functions are neither assigned nor supposed. The earliest occurrence of the idea of functionality is in connection with functions of real variables ; and then it is coextensive with the idea of dependence. Thus, if the value of X depends on that of x and on no other variable magnitude, it is customary to regard X as a function of x\ and there is usually an implication that X is derived from x by some series of operations^. A detailed knowledge of z determines x and y uniquely ; hence the values of u and v may be considered as known and therefore also w. Thus the value of w is dependent on that of z, and is independent of the values of variables unconnected with z; therefore, with the foregoing view of functionality, w is a function of z. It is, however, equally consistent with that view to regard w as a complex function of the two independent elements from which z is constituted ; and we are then led merely to the consideration of functions of two real independent variables with (possibly) imaginary coefficients. * These and other introductory parts of the subject are discussed in Chrystal's Text-book of Algebra and in Hobson's Treatise on Plane Trigonometry. They are also discussed at some length in the recently published translation, by G. L. Cathcart, of Harnack's Elements of the differential and integral calculus (Williams and Norgate, 1891), the second and the fourth books of which contain developments that should be consulted in special relation with the first few chapters of the present treatise. These books, together with Neumann's treatise.cited in the note on p. 5, will hereafter be cited by the names of their respective authors. t It is not important for the present purpose to keep in view such mathematical expressions as have intelligible meanings only when the independent variable is confined within limits. 6.] FUNCTIONAL DEPENDENCE 7 Both of these aspects of the dependence of w on z require that z be regarded as a composite quantity involving two independent elements which can be considered separately. Our purpose, however, is to regard z as the most general form of algebraical variable and therefore as an irresoluble entity ; so that, as this preliminary requirement in regard to z is unsatisfied, neither of the aspects can be adopted. 7. Suppose that w is regarded as a function of z in the sense that it can be constructed by definite operations on z regarded as an irresoluble magnitude, the quantities u and v arising subsequently to these operations by the separation of the real and the imaginary parts when z is replaced by x + iy. It is thereby assumed that one series of operations is sufficient for the simultaneous construction of u and v, instead of one series for u and another series for v as in the general case of a complex function in § 6. If this assumption be justified by the same forms resulting from the two different methods of construction, it follows that the two series of opera tions, which lead in the general case to u and to v, must be equivalent to the single series and must therefore be connected by conditions ; that is, u and v as functions of a; and y must have their functional forms related. We thus take u + iv — w = f(z) = f(x + iy) without any specification of the form of f. When this postulated equation is valid, we have dw dw dz ,. , . dw _ — . _ _ - I { 2/9 TTT _ dx dz dx J ^ ' dz' dw _ dw "dz _ .,,. . . dw frj = ~fad~y~V (Z) lfa' • dw 1 dw dw and therefore — = -— = —- ........................... (1) dx i dy dz equations from which the functional form has disappeared. Inserting the value of w, we have whence, after equating real and imaginary parts, dv _du du _ dv dx dy' dx dy" These are necessary relations between the functional forms of u and v. These relations are easily seen to be sufficient to ensure the required functionality. For, on taking w = ii + iv, the equations (2) at once lead to dw _ 1 dw dx i dy ' ,, , . dw .dw that is, to -- — \- 1 — - = 0, ox dy 8 RIEMANN'S [7. a linear partial differential equation of the first order. To obtain the most general solution, we form a subsidiary system dx _ dy _ dw T==T ==~0~* It possesses the integrals w, x + iy; and then from the known theory of such equations we infer that every quantity w satisfying the equation can be expressed as a function of x + iy, i.e., of z. The conditions (2) are thus proved to be sufficient, as well as necessary. 8. The preceding determination of the necessary and sufficient conditions of functional dependence is based upon the existence of a functional form ; and yet that form is not essential, for, as already remarked, it disappears from the equations of condition. Now the postulation of such a form is equivalent to an assumption that the function can be numerically calculated for each particular value of the independent variable, though the immediate expres sion of the assumption has disappeared in the present case. Experience of functions of real variables shews that it is often more convenient to use their properties than to possess their numerical values. This experience is confirmed by what has preceded. The essential conditions of functional dependence are the equations (1), and they express a property of the function w, viz., that the value of the ratio -r is the same as that of ~- , or, in other words, it is independent of the manner in which dz ultimately vanishes by the approach of the point z + dz to coincidence with the point z. We are thus led to an entirely different definition of functionality, viz. : A complex quantity w is a function of another complex quantity z, when they change together in such a manner that the value of -, is independent of the value of the differential element dz. This is Riemann's definition* ; we proceed to consider its significance. We have dw du + idv dz dx + idy /du .dv\ dx /du .dv\ du __ I __ I n _ I _ _____ I I __ L ^ __ I _ Y. _ ~~ \dx dxj dx + idy \dy dy/ dx + idy ' Let </> be the argument of dz ; then _ cos <£ + 1 sin </> * Ges. Werke, p. 5; a modified definition is adopted by him, ib., p. 81. 8.] DEFINITION OF A FUNCTION and therefore dw . (du .dv .du dv) „,. {du .dv .du dv I I I i n I I I 1 a—^4> I J I t In 7 — i i«i T" ^ • 5 • 57 f • a** i<5 Tfc ^~ ~r * « « «£ • [da; dx dy dy} (dx dx dy dy Since -j— is to be independent of the value of the differential element dz, dz it must be independent of <f> the argument of dz ; hence the coefficient of e-2*« in the preceding expression must vanish, which can happen only if du _dv dv _ du dx dy' dx dy " These are necessary conditions; they are evidently also sufficient to make ^— independent of the value of dz and therefore, by the definition, to secure that w is a function of z. By means of the conditions (2), we have dw _ du .dv _dw dz dx dx dx ' dw .du dv 1 dw and also — - = — i - — [_=_. dz dy dy i dy agreeing with the former equations (1) and immediately derivable from the present definition by noticing that dx and idy are possible forms of dz. It should be remarked that equations (2) are the conditions necessary and sufficient to ensure that each of the expressions udx — vdy and vdx + udy is a perfect differential — a result of great importance in many investigations in the region of mathematical physics. When the conditions (2) are expressed, as is sometimes convenient, in terms of derivatives with regard to the modulus of z, say r, and the argument of z, say 0, they take the new forms du_ldv dv _ Idu, ^ — ~ 57j > ^~ — ^TT. (^)- or r dv or r da We have so far assumed that the function has a differential coefficient — an assumption justified in the case of functions which ordinarily occur. But functions do occur which have different values in different regions of the .z-plane, and there is then a difficulty in regard to the quantity ,W at the boundaries of such regions ; and functions do occur which, though themselves definite in value in a given region, do not possess a differential coefficient at all points in that region. The consideration of such functions is not of substantial importance at present : it belongs to another part of our subject. 10 CONFORMAL [8. It must not be inferred that, because -j- is independent of the direction in which dz vanishes when w is a function of z, therefore -=- has only one value. The number of its values is dependent on the number of values of w : no one of its values is dependent on dz. A quantity, defined as a function by Riemann on the basis of this property, is sometimes* called an analytical function; but it seems pre ferable to reserve the term analytical in order that it may be associated hereafter (§ 34) with an additional quality of the functions. 9. The geometrical interpretation of complex variability leads to impor tant results when applied to two variables w and z which are functionally related. Let P and p be two points in different planes, or in different parts of the same plane, representing w and z respectively; and suppose that P and p are at a finite distance from the points (if any) which cause discontinuity in the relationship. Let q and r be any two other points, z + dz and z + 8z, in the immediate vicinity of p ; and let Q and E be the corresponding points, w + dw and w + &w, in the immediate vicinity of P. Then dw j ^ dw ? dw = ^r- dz. bw = -r— of, dz dz the value of ~ being the same for both equations, because, as w is a function dz of z, that quantity is independent of the differential element of z. Hence 8w _ Bz dw dz' on the ground that , is neither zero nor infinite at z, which is assumed not CL2 to be a point of discontinuity in the relationship. Expressing all the differ ential elements in terms of their moduli and arguments, let dz = a-eei, dw — rje^1, Sz = oV'*, 8w = i)<$\ and let these values be substituted in the foregoing relation ; then 77' tr tj a $-$ = &-&. Hence the triangles QPR and qpr are similar to one another, though not necessarily similarly situated. Moreover the directions originally chosen for pq and pr are quite arbitrary. Thus it appears that a functional relation * Harnack, § 84. 1 11 V - (<M\* a. i^v • I <a I — I "5 / ' \ "> $a?/ \oyj \dy 9.] REPRESENTATION OF PLANES 11 between two complex variables establishes the similarity of the corresponding infinitesimal elements of those parts of two planes which are in the immediate vicinity of the points representing the two variables. The magnification of the w-plane relative to the ^-plane at the corre sponding points P and p is the ratio of two corresponding infinitesimal lengths, say of QP and qp. This is the modulus of -^— ; if it be denoted by m, we have 2 _ dw 2 dz _ du dv du dv dx dy dy dx ' Evidently the quantity m, in general, depends on the variables and therefore it changes from one point to another ; hence the functional relation between w and z does not, in general, establish similarity of finite parts of the two planes corresponding to one another through the relation. It is easy to prove that w = az + b, where a and b are constants, is the only relation which establishes similarity of finite parts ; and that, with this relation, a must be a real constant in order that the similar parts may be similarly situated. If u + iv = w = <}> (z), the curves u = constant and v = constant cut at right angles; a special case of the proposition that, if <£ (x + iy) = u + v^, where A, is a real constant and u, v are real, then u= constant and v= constant cut at an angle X. The process, which establishes the infinitesimal similarity of two planes by means of a functional relation between the variables of the planes, may be called the conformal representation of one plane on another*. The discussion of detailed questions connected with the conformal representation is deferred until the later part of the treatise, principally in order to group all such investigations together ; but the first of the two chapters, devoted to it, need not be deferred so late and an immediate reading of some portion of it will tend to simplify many of the explanations relative to functional relations as they occur in the early chapters of this treatise. 10. The analytical conditions of functionality, under either of the adopted definitions, are the equations (2). From them it at once follows that 8^ + ty* = ' * By Gauss (Ges. Werke, t. iv, p. 262) it was styled conforme Abbildung, the name universally adopted by German mathematicians. The French title is representation conforme ; and, in England, Cayley has used orthomorphosis or ortliomorphic transformation. 12 CONDITIONS OF FUNCTIONAL DEPENDENCE [10. so that neither the real nor the imaginary part of a complex function can be arbitrarily assumed. If either part be given, the other can be deduced ; for example, let u be given ; then we have 7 j j dv = ^-dx + — dy dx dy du , du j = -=-dx+~-dy, dy ox ' and therefore, except as to an additive constant, the value of v is [i 9w 7 du -, \ - — dx + 5- dy I . A dy ax ° I In particular, when u is an integral function, it can be resolved into the sum of homogeneous parts MI + w2 + w3 + . . . ; and then, again except as to an additive constant, v can similarly be expressed in the form Vl + V2 + V3 + ---- It is easy to prove that dum dum ™>» = y-te-*-ty> by means of which the value of v can be obtained. The case, when u is homogeneous of zero dimensions, presents no difficulty ; for we then have v = c-a\ogr, =c-/f£ where a, 6, c are constants. Similarly for other special cases; and, in the most general case, only a quadrature is necessary. The tests of functional dependence of one complex on another are of effective importance in the case when the supposed dependent complex arises in the form u + iv, where u and v are real; the tests are, of course, superfluous when w is explicitly given as a function of z. When w does arise in the form u + iv and satisfies the conditions of functionality, perhaps the simplest method (other than by inspection) of obtaining the explicit expression in terms of z is to substitute z — iy for x in u + iv ; the simplified result must be a function of z alone. 11. Conversely, when w is explicitly given as a function of z and it is divided into its real and its imaginary parts, these parts individually satisfy the foregoing conditions attaching to u and v. Thus logr, where r is the distance of a point z from a point a, is the real part of log (z — a) and therefore satisfies the equation 11.] EXAMPLE OF RIEMANN S DEFINITION 13 Again, <f>, the angular coordinate of z relative to the same point a, is the real part of — i log (z — a) and satisfies the same equation : the more usual form of <£ being tan"1 {(y — y0)/(® — %o)}> where a = x0 + iy0. Again, if a point z be distant r from a and r' from b, then log (r/r'\ being the real part of log {(z — a)l(z — b)\, is a solution of the same equation. The following example, the result of which will be useful subsequently*, uses the property that the value of the derivative is independent of the differential element. z-c Consider a function u + iv = w = log where c' is the inverse of c with regard to a circle centre the origin 0 and radius R. Then z-c * V :— r> z-c and the curves u = constant are circles. Let W- • (fig. 4) Oc = r, xOc = a so that c = reat, c'= — eal; then if Fig. 4. the values of X for points in the interior of the circle of radius R vary from zero, when circle u = constant is the point c, to unity, when the circle u = constant is the circle of radius R. Let the point K ( = 6eal) be the centre of the circle determined by a value of X, and let its radius be p ( = %MN}. Then since cM r ,. cN we have whence r+p-d r d + P~r — Vp-B Q-p r r P = Now if dn be an element of the normal drawn inwards at z to the circle NzM, we have dz = dx+idy= — dn . cos ^ - idn . sin ^ --«*<*», where ^ ( = zKx'} is the argument of z relative to the centre of the circle. Hence, since dw 1 1 we have But so that and , ., , du .dv and therefore -=- + i -j- = dn dn __ _ dz z — c z-c'1 du .dv dw dn .dv dw /I 1 \ ty dn dn \z — c' z — c) e^ - Reai) • J> _ 1 _ 1 !_ I /i! ~\ ^^ 7? *1^ X ff */^ \ A7*6 — J\G ./t6 i * In § 217, in connection with the investigations of Schwarz, by whom the result is stated, Ges. Werke, t. ii, p. 183. 14 DEFINITIONS [11. Hence, equating the real parts, it follows that du (_R2-r2A2)2 dn ~ \R(R*- r2) {E2 - 2Rr\ cos (^ - Q) + XV2} ' the differential element dn being drawn inwards from the circumference of the circle. The application of this method is evidently effective when the curves u = constant, arising from a functional expression of w in terms of z, are a family of non-intersecting algebraical curves. 12. As the tests which are sufficient and necessary to ensure that a complex quantity is a function of z have been given, we shall assume that all complex quantities dealt with are functions of the complex variable (§§ 6, 7). Their characteristic properties, their classification, and some of the simpler applications will be considered in the succeeding chapters. Some initial definitions and explanations will now be given. (i). It has been assumed that the function considered has a differential coefficient, that is, that the rate of variation of the function in any direction is independent of that direction by being independent of the mode of change of the variable. We have already decided (§ 8) not to use the term analytical for such a function. It is often called monogenic, when it is necessary to assign a specific name ; but for the most part we shall omit the name, the property being tacitly assumed*. We can at once prove from the definition that, when the derivative / dw\ •.-.'•, if- c <-• v dw Idw w, = -p- exists, it is itselt a Junction, .bor w-, =-=— = - =— are equations \ dz ) dx i dy which, when satisfied, ensure the existence of w^ ; hence 1 dw-! _ 1 3 (dw\ i dy i dy \d% ) _ d_ (I dw\ dx \i dyj _dw1 = l)x ' shewing, as in § 8, that the derivative ~ is independent of the direction in CL2 which dz vanishes. Hence wl is a function of z. Similarly for all the derivatives in succession. (ii). Since the functional dependence of a complex is ensured only if the value of the derivative of that complex be independent of the manner in which the point z + dz approaches to coincidence with z, a question naturally * This is in fact done by Biemann, who calls such a dependent complex simply a function. Weierstrass, however, has proved (§ 85) that the idea of a monogenic function of a complex variable and the idea of dependence expressible by arithmetical operations are not coextensive. The definition is thus necessary; but the practice indicated in the text will be adopted, as non- monogenic functions will be of relatively rare occurrence. 12.] DEFINITIONS 15 suggests itself as to the effect on the character of the function that may be caused by the manner in which the variable itself has come to the value of z. If a function have only one value for each given value of the variable, whatever be the manner in which the variable has come to that value, the function is called uniform*. Hence two different paths from a point a to a point z give at z the same value for any uniform function ; and a closed curve, beginning at any point and completely described by the ^-variable, will lead to the initial value of w, the corresponding w-curve being closed, if z have passed through no point which makes w infinite. The simplest class of uniform functions is constituted by algebraical rational functions. (iii). If a function have more than one value for any given value of the variable, or if its value can be changed by modifying the path in which the variable reaches that given value, the function is called multiform-]'. Characteristics of curves, which are graphs of multiform functions corre sponding to a 2-curve, will hereafter be discussed. One of the simplest classes of multiform functions is constituted by algebraical irrational functions. (iv). A multiform function has a number of different values for the same value of z, and these values vary with z : the aggregate of the variations of any one of the values is called a branch of the function. Although the function is multiform for unrestricted variation of the variable, it often happens that a branch is uniform when the variable is restricted to particular regions in the plane. (v). A point in the plane, at which two or more branches of a multiform function assume the same value, is called a branch-point^ of the function; the relations of the branches in the immediate vicinity of a branch-point will hereafter be discussed. (vi). A function which is monogenic, uniform and continuous over any part of the ^-plane is called holomorphic § over that part of the plane. When •a function is called holomorphic without any limitation, the usual implication is that the character is preserved over the whole of the plane which is not at infinity. The simplest example of a holomorphic function is a rational integral algebraical polynomial. * Also monodromic, or monotropic; with German writers the title is eindeutig, occasionally, einandrig. t Also polytropic ; with German writers the title is mchrdeittig. J Also critical point, which, however, is sometimes used to include all special points of a function ; with German writers the title is Verziveigungspunkt, and sometimes Windungspunkt. French writers use point de ramification, and Italians punto di giramento and punto di diramazione. § Also synectic. 16 EXAMPLES ILLUSTRATING [12. (vii). A root (or a zero) of a function is a value of the variable for which the function vanishes. The simplest case of occurrence of roots is in a rational integral alge braical function, various theorems relating to which (e.g., the number of roots included within a given contour) will be found in treatises on the theory of equations. (viii). The infinities of a function are the points at which the value of the function is infinite. Among them, the simplest are the poles* of the function, a pole being an infinity such that in its immediate vicinity the reciprocal of the function is holomorphic. Infinities other than poles (and also the poles) are called the singular points of the function : their classification must be deferred until after the discussion of properties of functions. (ix). A function which is monogenic, uniform and, except at poles, continuous, is called a meromorphic function f. The simplest example is a rational algebraical fraction. 13. The following functions give illustrations of some of the preceding definitions. (a) In the case of a meromorphic function F(z) 111 — — * — - /<*)' where F and / are rational algebraical functions without a common factor, the roots are the roots of F (z) and the poles are the roots of f (z). Moreover, according as the degree of F is greater or is less than that of f,z = vo is a pole or a zero of w. (b) If w be a polynomial of order n, then each simple root of w is a branch-point and a zero of wm, where m is a positive integer ; z = oo is a pole of w; and z= oo is a pole but not a branch-point or is an infinity (though not a pole) and a branch-point of w$ according as n is even or odd. (c) In the case of the function 1 w- sn- z (the notation being that of Jacobian elliptic functions), the zeros are given by z for all positive and negative integral values of m and of m'. If we take - = iK' + 2mK + Zm'iK' -f £ z * Also polar discontinuities ; also (§ 32) accidental singularities. t Sometimes rey-nlar, but this term will be reserved for the description of another property of functions. 13.] THE DEFINITIONS 17 where £ may be restricted to values that are not large, then w = (- l)m &sn£ so that, in the neighbourhood of a zero, w behaves like a holomorphic function. There is evidently a doubly-infinite system of zeros: they are distinct from one another except at the origin, where an infinite number practically coincide. The infinities of w are given by for all positive and negative integral values of n and of n'. If we take - = 2nK + Zn'iK' + £ 2! then - = (-l)"sn£ w so that, in the immediate vicinity of f=0, - is a holomorphic function. Hence f = 0 is a pole of w. There is thus evidently a doubly-infinite system of poles ; they are distinct from one another except at the origin, where an infinite number practically coincide. But the origin is not a pole; the function, in fact, is there not determinate, for it has an infinite number of zeros and an infinite number of infinities, and the variations of value are not necessarily exhausted. For the function — j , the origin is a point which will hereafter be called sn- z an essential singularity. F. CHAPTER II. INTEGRATION OF UNIFORM FUNCTIONS. 14. THE definition of an integral, that is adopted when the variables are complex, is the natural generalisation of that definition for real variables in which it is regarded as the limit of the sum of an infinite number of infinitesimally small terms. It is as follows : — Let a and z be any two points in the plane ; and let them be connected by a curve of specified form, which is to be the path of variation of the independent variable. Let f(z) denote any function of 0; if any infinity of f(z) lie in the vicinity of the curve, the line of the curve will be chosen so as not to pass through that infinity. On the curve, let any number of points z^ z2,..., zn in succession be taken between a and z ; then, if the sum (z, - a)f (a) + (z, - z,} f (z,) + ... + (z- zn)f(zn} have a limit, when n is indefinitely increased so that the infinitely numerous points are in indefinitely close succession along the whole of the curve from a to z, that limit is called the integral of / (z) between a and z. It is denoted, as in the case of real variables, by f(z)dz. The limit, as the value of the integral, is associated with a particular curve : in order that the integral may have a definite value, the curve (called the path of integration) must, in the first instance, be specified*. The integral of any function whatever may not be assumed to depend in general only upon the limits. 15. Some inferences can be made from the definition. (I.) The integral along any path from a to z passing through a point £ is the sum of the integrals from a to £ and from \ to z along the same path. * This specification is tacitly supplied when the variables are real : the variable point moves along the axis of x. 15.] INTEGRATION 19 Analytically, this is expressed by the equation P / (*) dz = I V (*) dz + I V (*) <fc, ^ a J a J f the paths on the right-hand side combining to form the path on the left. (II.) When the path is described in the reverse direction, the sign of the integral is changed : that is, the curve of variation between a and z being the same. (III.) The integral of the sum of a finite number of terms is equal to the sum of the integrals of the separate terms, the path of integration being the same for all. (IV.) If a function f (z) be finite and continuous along any finite line between two points a and z, the integral \ f(z)dz is finite. J a Let 7 denote the integral, so that we have I as the limit of r=0 hence |/| = limit of Because f(z} is finite and continuous, its modulus is finite and therefore must have a superior limit, say M, for points on the line. Thus 80 that I/I < limit of r+1 <MS, where 8 is the finite length of the path of integration. Hence the modulus of the integral is finite ; the integral itself is therefore finite. No limitation has been assigned to the path, except finiteness in length ; the proposition is still true when the curve is a closed curve of finite length. Hermite and Darboux have given an expression for the integral which leads to the same result. We have as above f(z)\ dz\, where 6 is a real positive quantity less than unity. The last integral involves 2—2 20 THEOREMS [15. only real variables; hence* for some point £ lying between a and z, we have f J a so that l/| = fl9f|/(!)|. It therefore follows that there is some argument a such that, if X = Be10-, This form proves the finiteness of the integral ; and the result is the generalisation f to complex variables of the theorem just quoted for real variables. (V.) When a, function is expressed in the form of a series, which converges uniformly and unconditionally, the integral of the function along any path of finite length is the sum of the integrals of the terms of the series along the same path, provided that path lies within the circle of convergence of the series : — a result, which is an extension of (III.) above. Let M0 + MI + u.2 + . . . be the converging series ; take / (z) = U0 + M! + . . . + Un + R, where \R\ can be made infinitesimally small with indefinite increase of n, because the series converges uniformly and unconditionally. Then by (III.), or immediately from the definition of the integral, we have rz rs rz rz re f(z)dz= I u0dz + ^dz + . .. + I undz + 1 Rdz, J a J a J a J a J a the path of integration being the same for all the integrals. Hence, if re n re (S) = I f (z) dz — 2 I umdz, J a m=oJ a ft we have © = I Rdz. ft = I J a Let R be the greatest value of \R\ for points in the path of integration from a to z, and let 8 be the length of this path, so that 8 is finite ; then, by (IV.), \®\<SR. Now 8 is finite ; and, as n is increased indefinitely, the quantity R tends towards zero as a limit for all points within the circle of convergence and therefore for all points on the path of integration provided that the path lie within the circle of convergence. When this proviso is satisfied, |@| becomes infinitesimally small and therefore also ® becomes infinitesimally small with * Todhunter's Integral Calculus (4th ed.), § 40; Williamson's Integral Calculus, (Gth ed.), § 96. t Hermite, Cours d la faculte dcs sciences de Paris (46mc ed., 1891), p. 59, where the reference to Darboux is given. 15.] ON INTEGRATION 21 indefinite increase of n. Hence, under the conditions stated in the enuncia tion, we have rs oo r% f(z)dz- 2 I umdz = 0, J a m^QJ a which proves the proposition. 16. The following lemma* is of fundamental importance. Let any region of the plane, on which the ^-variable is represented, be bounded by one or more simple^ curves which do not meet one another: each curve that lies entirely in the finite part of the plane will be considered to be a closed curve. If ' p and q be any two functions of cc and y, which, for all points within the region or along its boundary, are uniform, finite and continuous, then the integral fffdq dp\j , 1 1 a - a dxdy, JJ \dx dyj extended over the whole area of the region, is equal to the integral f(pdx + qdy), taken in a positive direction round the whole boundary of the region. (As the proof of the proposition does not depend on any special form of region, we shall take the area to be (fig. 5) that which is included by the curve QiPiQs'Pa' and excluded by P^Qz'PsQs and excluded by P/P2. The positive directions of description of the curves are indicated by the arrows ; and for integration in the area the positive directions are those of increas ing a; and increasing y.) AB Fig. 5. * It is proved by Eiemann, Ges. Werke, p. 12, and is made by him (as also by Cauchy) the basis of certain theorems relating to functions of complex variables. t A curve is called simple, if it have no multiple points. The aim, in constituting the boundary from such curves is to prevent the superfluous complexity that arises from duplication of area on the plane. If, in any particular case, multiple points existed, the method of meeting the difficulty would be to take each simple loop as a boundary. 22 FUNDAMENTAL THEOREM [16. First, suppose that both p and q are real. Then, integrating with regard to x, we have * where the brackets imply that the limits are to be introduced. When the limits are introduced along a parallel GQ^... to the axis of x, then, since CQiQi'. • • gives the direction of integration, we have [qdy] = - qjdyj. + qi'dt/i - q.2dy2 + q-2'dy2' - q3dy3 + q»dy9', where the various differential elements are the projections on the axis of y of the various elements of the boundary at points along GQiQJ.... Now when integration is taken in the positive direction round the whole boundary, the part of / qdy arising from the elements of the boundary at the points on CQjQ/... is the foregoing sum. For at Q3' it is qa'dy3 because the positive element dy9, which is equal to CD, is in the positive direction of boundary integration; at Q3 it is —q3dys because the positive element dy3, also equal to CD. is in the negative direction of boundary integration ; at Qz it is q2'dy2', for similar reasons ; at Q.2 it is — q2dya, for similar reasons ; and so on. Hence corresponding to parallels through C and D to the axis of x, is equal to the part of fqdy taken along the boundary in the positive direction for all the elements of the boundary that lie between those parallels. Then when we integrate for all the elements CD by forming f[qdy], an equivalent is given by the aggregate of all the parts of fqdy taken in the positive direction round the whole boundary ; and therefore on the suppositions stated in the enunciation. Again, integrating with regard to y, we have when the limits are introduced along a parallel RP^P^. . . to the axis of y : the various differential elements are the projections on the axis of x of the various elements of the boundary at points along SPjP/.... It is proved, in the same way as before, that the part of - jpdx arising from the positively-described elements of the boundary at the points on BP^'... is the foregoing sum. At P3 the part of fpdac is - p3'dx3, because the positive element dx3, which is equal to AB, is in the negative direction * It is in this integration, and in the corresponding integration for p, that the properties of the function q are assumed : any deviation from uniformity, finiteness or continuity within the region of integration would render necessary some equation different from the one given in the text. 16.] IN INTEGRATION 23 of boundary integration ; at P3 it is p3dx3, because the positive element dx3, also equal to AB, is in the positive direction of boundary integration; and so on for the other terms. Hence - [pdas], corresponding to parallels through A and B to the axis of y, is equal to the part of fpdx taken along the boundary in the positive direction for all the elements of the boundary that lie between those parallels. Hence integrating for all the elements AB, we have as before [[dp j j , j ~ dxdy = — I pax, JJdy and therefore II U ?r ) dxdy=f(pdx + qdy). Secondly, suppose that p and q are complex. When they are resolved into real and imaginary parts, in the forms p' + ip" and q' + iq" respectively, then the conditions as to uniformity, finiteness and continuity, which apply to p and q, apply also to p', q', p", q". Hence and ~ - - dxdy = j(p"dx + q"dy), and therefore 1 1 [ 2* _ J9 j dxdy = J(pdx + qdy} JJ \ox oy/ which proves the proposition. No restriction on the properties of the functions p and q at points that lie without the region is imposed by the proposition. They may have infinities outside, they may cease to be continuous at outside points or they may have branch-points outside ; but so long as they are finite and continuous everywhere inside, and in passing from one point to another always acquire at that other the same value whatever be the path of passage in the region, that is, so long as they are uniform in the region, the lemma is valid. 17. The following theorem due to Cauchy* can now be proved : _ If a function f(z) be holomorphic throughout any region of the z-plane, then the integral ff(z) dz, taken round the whole boundary of that region, is zero. We apply the preceding result by assuming p=f(z\ q = ip = if(z); owing to the character of f(z), these suppositions are consistent with the * For an account of the gradual development of the theory and, in particular, for a statement of Cauchy's contributions to the theory (with references), see Casorati, Teorica delle funzioni di variabili complcsse, pp. 64-90, 102-106. The general theory of functions, as developed by Briot and Bouquet in their treatise Theoric des fonctiom ellipUques, is based upon Cauchy's method. 24 INTEGRATION OF [17. conditions under which the lemma is valid. Since p is a function of z, we have, at every point of the region, dp _ I dp das i dy ' and therefore, in the present case, dq _ . dp _ dp das doc dy ' There is no discontinuity or infinity of p or q within the region ; hence the integral being extended over the region. Hence also !(pdx + qdy) = 0, A^ ^/ when the integral is taken round the whole boundary of the region. But pdx + qdy = pdx + ipdy — pdz =f(z)dz, and therefore //(X) dz = 0, the integral being taken round the whole boundary of the region within which f(z) is holomorphic. It should be noted that the theorem requires no limitation on the cha racter of/(^) for points z that are not included in the region. Some important propositions can be derived by means of the theorem, as follows. 18. When a function f (z) is holomorphic over any continuous region rz of the plane, the integral I f(z)dz is a holomorphic function of 2 provided the J a points z and a as well as the whole path of integration lie within that region. The general definition (§ 14) of an integral is associated with a specified path of integration. In order to prove that the integral is a holomorphic function of z, it will be necessary to prove (i) that the integral acquires the same value in whatever way the point z is attained, that is, that the value is independent of the path of integration, (ii) that it is finite, (iii) that it is continuous, and (iv) that it is monogenic. Let two paths ayz and afiz between a and z be drawn (fig. 6) in the continuous region of the plane within which f(z) is holomorphic. The line ayzfia is a contour over the area of which / (z} is holomorphic ; and therefore ff(z) dz vanishes when the integral is taken along ayzfta. Dividing the integral into two parts and implying by Zy, Zp that the point z has been reached by the paths a" a<yz, a{3z respectively, we have Fig. 6. 18.] HOLOMOEPHIC FUNCTIONS 25 and therefore */ (z) dz = - f (z) dz J a J Zg -?/*/(*)* J a Thus the value of the integral is independent of the way in which z has FZ acquired its value ; and therefore I f(z) dz is uniform in the region. Denote it by F(z). Secondly, f(z) is finite for all points in the region and, after the result of § 17, we naturally consider only such paths between a and z as are finite in length, the distance between a and z being finite; hence (§ 15, IV.) the integral F (z} is finite for all points z in the region. Thirdly, let z' (= z 4- 82) be a point infinitesimally near to z ; and consider I f(z) dz. By what has just been proved, the path from a to z' can be taken J d aftzz' ; therefore (*/(*) dz = [/(z) dz + lZf(z) dz J * J a J z fz+8z rz rz+Sz or f(z}dz- \ f(z)dz=\ f(z)dz, J a J a J z fz+Sz 80 that F(z + Sz) - F(z) = f(z} dz. J 2 Now at points in the infinitesimal line from z to z' , the value of the continuous function f(z) differs only by an infinitesimal quantity from its value at z ; hence the right-hand side is where e| is an infinitesimal quantity vanishing with ck It therefore follows that is an infinitesimal quantity with a modulus of the same order of small quantities as \Sz\. Hence F (z) is continuous for points z in the region. Lastly, we have and therefore F(z + Sz)-F(z) 82 has a limit when Sz vanishes; and this limit, f(z), is independent of the way in which 8z vanishes. Hence F (z) has a differential coefficient ; the integral is monogenic for points z in the region. 26 INTEGRATION OF [18. Hence F (z), which is equal to * f(z)d*t is uniform, finite, continuous and monogenic; it is therefore a holomorphic function of z. As in § 16 for the functions p and q, so here for f(z), no restriction is placed on properties of / (z) at points that do not lie within the region; so that elsewhere it may have infinities, or discontinuities or branch points. The properties, essential to secure the validity of the proposition, are (i) that no infinities or discontinuities lie within the region, and (ii) that the same value of f(z) is acquired by whatever path in the continuous region the variable reaches its position z. COROLLARY. No change is caused in the value of the integral of a holomorphic function between two points when the path of integration between the points is deformed in any manner, provided only that, during the defor mation, no part of the path passes outside the boundary of the region within which the function is holomorphic. This result is of importance, because it permits special forms of the path of integration without affecting the value of the integral. 19. When a function f(z) is holomorphic over a part of the plane bounded by two simple curves (one lying within the other), equal values of ff(z) dz are obtained by integrating round each of the curves in a direction, which — relative to the area enclosed by each — is positive. The ring-formed portion of the plane (fig. 1, p. 3) which lies between the two curves being a region over which f(z) is holomorphic, the integral ff(z) dz taken in the positive sense round the whole of the boundary of the included portion is zero. The integral consists of two parts : first, that round the outer boundary the positive sense of which is DEF', and second, that round the inner boundary the positive sense of which for the portion of area between ABC and DEF is ACE. Denoting the value of ff(z)dz round DEF by (DEF), and similarly for the other, we have (ACB) + (DEF) = 0. The direction of an integral can be reversed if its sign be changed, so that (ACB) = - (ABC) ; and therefore (ABC) = (DEF). But (ABC) is the integral ff(z)dz taken round ABC, that is, round the curve in a direction which, relative to the area enclosed by it, is positive. The proposition is therefore proved. The remarks made in the preceding case as to the freedom from limitations on the character of the function outside the portion are valid also in this case. 19.] HOLOMORPHIC FUNCTIONS 27 COROLLARY I. When the integral of a function is taken round the whole of any simple curve in the plane, no change is caused in its value by continuously deforming the curve into any other simple curve provided that the function is holomorphic over the part of the plane in which the deformation is effected. COROLLARY II. When a function f (z) is holomorphic over a continuous portion of a plane bounded by any number of simple non-intersecting curves, all but one of which are external to one another and the remaining one of which encloses them all, the value of the integral jf(z) dz taken positively round the single external curve is equal to the sum of the values taken round each of the other curves in a direction which is positive relative to the area enclosed by it. These corollaries are of importance in finding the value of the integral of a meromorphic function round a curve which encloses one or more of the poles. The fundamental theorem for such integrals, also due to Cauchy, is the following. 20. Let f(z) denote a function which is holomorphic over any region in the z-plane and let a denote any point within that region, which is not a zero °ff(2); then ., , 1 f/0) , f(a) = ^— • *-*-* az> 2vnJ z-a the integral being taken positively round the whole boundary of the region. With a as centre and a very small radius p, describe a circle G, which will be assumed to lie wholly within the region; this assumption is justifiable because the point a lies within the region. Because f (z) is holomorphic over the assigned region, the f unction f(z)l(z — a) is holomorphic over the whole of the region excluded by the small circle C. Hence, by Corollary II. of § 19, we have z-a the notation implying that the integrations are taken round the whole boundary B and round the circumference of G respectively. For points on the circle C, let z — a = peei, so that 9 is the variable for the circumference and its range is from 0 to 2?r ; then we have dz z — a = id6. Along the circle f(z)=f (a + peei) ; the quantity p is very small and / is finite and continuous over the whole of the region so that f(a + peei) differs from /(«) only by a quantity which vanishes with p. Let this difference be e, which is a continuous small quantity; then |ej is a small quantity which, for every point on the circumference of C, vanishes with p. Then 28 INTEGRATION OF [20. " edO. o If E denote the value of the integral on the right-hand side, and 77 the greatest value of the modulus of e along the circle, then, as in § 15, /•2ir i E < I e d6 f Now let the radius of the circle diminish to zero: then 77 also diminishes to zero and therefore E , necessarily positive, becomes less than any finite quantity however small, that is, E is itself zero; and thus we have z — a which proves the theorem. This result is the simplest case of the integral of a meromorphic f(z} function. The subject of integration is — — , a function which is monogenic and uniform throughout the region and which, everywhere except at z = a, is finite and continuous ; moreover, z = a is a pole, because in the immediate Z ~~~ CL vicinity of a the reciprocal of the subject of integration, viz. ^-rr > i-B h°l°- morphic. The theorem may therefore be expressed as follows : If g (z) be a meromorphic function, which in the vicinity of a can be f(z} expressed in the form J where f(a) is not zero and which at all other Z — CL points in a region enclosing a is holomorphic, then - — . fg (z) dz = limit of (z — a)g (z) when z — a, the integral being taken round a curve in the region enclosing the point a. The pole a of the function g (z) is said to be simple, or of the first order, or of multiplicity unity. Corollary. The more general case of a meromorphic function with a finite number of poles can easily be deduced. Let these be a1}..., an each assumed to be simple ; and let G (z) = (z- a,) (z - aa). ..(z - an). 20.] MEROMORPHIC FUNCTIONS 29 Let f(z) be a holomorphic function within a region of the 2-plane bounded by a simple contour enclosing the n points a1} a»,...an, no one of which is a zero off(z). Then since f(z) » 1 f(z) we have j^~( = S „,, . -^-- . 6r (#) r=i Or (ar) z — ar w ^ f u f/(*),j 3 ! f/(*) ,7 We therefore have "L , ' dz = 2< >.. , . I dz, J &(*) r=iCr (ar)J 2-ar each integral being taken round the boundary. But the preceding proposition gives because f(z) is holomorphic over the whole region included in the contour ; and therefore the integral on the left-hand side being taken in the positive direction*. The result just obtained expresses the integral of the meromorphic function round a contour which includes a finite number of its simple poles. It can be otherwise obtained by means of Corollary II. of § 19, by adopting a process similar to that adopted above, viz., by making each of the curves in the Corollary quoted small circles round the points Oj,..., an with ultimately vanishing radii. 21. The preceding theorems have sufficed to evaluate the integral of a function with a number of simple poles : we now proceed to obtain further theorems, which can be used among other purposes to evaluate the integral of a function with poles of order higher than the first. We still consider a function f(z) which is holomorphic within a given region. Then, if a be a point within the region which is not a zero of f(z), we have z - a the point a being neither on the boundary nor within an infinitesimal distance of it. Let a + Sa be any other point within the region ; then dz, z — a — 8a * We shall for the future assume that, if no direction for a complete integral be specified, the positive direction is taken. 30 and therefore PROPERTIES OF [21. iff, 8a f(z)dz, t J ((* - a)2 (z - of (z-a -Sa)j the integral being in every case taken round the boundary. Since f(z) is monogenic, the definition of /'(a), the first derivative of /(a), gives /'(a) as the limit of f(a + Ba)-f(a) Ba when Ba ultimately vanishes ; hence we may take where a is a quantity which vanishes with Ba and is therefore such that \ a \ also vanishes with Ba. Hence dividing out by Sa and transposing, we have As yet, there is no limitation on the value of Sa ; we now proceed to a limit by making a + Ba approach to coincidence with a, viz., by making Ba ultimately vanish. Taking moduli of each of the members of the last equation, we have (a) _ i f J(* 2in j (z - o _„ + ««. (z — a)2 (z — a — Ba) 27T dz Let the greatest modulus of -. ~ =r—. for points z along the (j — a)2 (z — a — Ba) boundary be M, which is a finite quantity on account of the conditions applying to f(z) and the fact that the points a and a + Ba are not infinitesimally near the boundary. Then, by § 15, t dz '0-a)2 (z-a-Ba) <MS, where 8 is the whole length of the boundary, a finite quantity. Hence 1 f f(z} , , |8a| dz c ITT 21.] HOLOMORPHIC FUNCTIONS 31 When we proceed to the limit in which Sa vanishes, we have Ba = 0 and |o-| = 0, ultimately; hence the modulus on the left-hand side ultimately vanishes and therefore the quantity to which that modulus belongs is itself zero, that is, , (z — of so that / (a) = —-. !/-^~n dz. ZTTI )(z- of This theorem evidently corresponds in complex variables to the well-known theorem of differentiation with respect to a constant under the integral sign when all the quantities concerned are real. Proceeding in the same way, we can prove that / (a + &*)-/ (a) _ 2!_ f /(*) Ba ~2Trij(z-af where 6 is a small quantity which vanishes with Ba. Moreover the integral on the right-hand side is finite, for the subject of integration is everywhere finite along the path of integration which itself is of finite length. Hence, first, a small change in the independent variable leads to a change of the same order of small quantities in the value of the function f (a), which shews that f (a) is a continuous function. Secondly, denoting &*) -/(a) by &/'(«), we have the limiting value of -— *— - equal to the integral on the right-hand side when Sa vanishes, that is, the derivative of f (a) has a value independent of the form of 8a and therefore /' (a) is monogenic. Denoting this derivative by /"(a), we have J (z — a)3 Thirdly, the function f (a) is uniform ; for it is the limit of the value of — - -- x-- — J-\J and both /(a) and /(a + Sa) are uniform. Lastly, it is finite; for (S 15) it is the value of the integral - — . l.^—^dz, in which 2?n J (z — af the length of the path is finite and the subject of integration is finite at every point of the path. Hence f (a) is continuous, monogenic, uniform, and finite throughout the whole of the region in which f (z) has these properties: it is a holomorphic function. Hence : — When a function is holomorphic in any region of the plane bounded 32 PROPERTIES OF [21. by a simple curve, its derivative is also holomorphic within that region. And, by repeated application of this theorem : — When a function is holomorphic in any region of the plane bounded by a simple curve, it has an unlimited number of successive derivatives each of which is- holomorphic within the region. All these properties have been shewn to depend simply upon the holo morphic character of the fundamental function ; but the inferences relating to the derivatives have been proved only for points within the region and not for points on the boundary. If the foregoing methods be used to prove them for points on the boundary, they require that a consecutive point shall be taken in any direction ; in the absence of knowledge about the fundamental function for points outside (even though just outside) no inferences can be justifiably drawn. An illustration of this statement is furnished by the hypergeometric series which, together with all its derivatives, is holomorphic within a circle of radius unity and centre the origin ; and the series converges unconditionally everywhere on the circumference, provided 7 > a. + /3. But the corresponding condition for convergence on the circumference ceases to be satisfied for some one of the derivatives and for all which succeed it : as such functions do not then converge unconditionally, the circumference of the circle must be excluded from the region within which the derivatives are holomorphic. 22. Expressions for the first and the second derivatives have been obtained. By a process similar to that which gives the value of f (a), the derivative of order n is obtainable in the form n ' f f (z\ /<») (a) = — . I, ' dz, J w 2wt J (z - a)n+l the integral being taken round the whole boundary of the region or round any curves which arise from deformation of the boundary, provided that no point of the curves in the final or any intermediate form is indefinitely near to a. In the case when the curve of integration is a circle, no point of which circle may lie outside the boundary of the region, we have a modified form fcr /*'(•> For points along the circumference of the circle with centre a and radius r, let z — a = reei, dz so that as before — = idO : z — a then 0 and 2?r being taken as the limits of 0, we have 22.] HOLOMORPHIC FUNCTIONS 33 Let M be the greatest value of the modulus of f (z) for points on the circumference (or, as it may be convenient to consider, of points on or within the circumference) : then \f(n)(a)\<~ e-nei\\f(a i / \ / 1 ^ 27ryw * - nl < M Now, let there be a function <£ (s) defined by the equation M — a which can evidently be expanded in a series of ascending powers of z — a that converges within the circle. The series is - [dnd> (z)~\ , M Hence —!L±J =n\ — [ d*» ]z=a *>• so that, if the value of the nth derivative of $(z), when z = a, be denoted by <£<n> (a), we have |/»(a)| «p>(a). These results can be extended to functions of more than one variable : the proof is similar to the foregoing proof. When the variables are two, say z and z', the results may be stated as follows : — ^ For all points z within a given simple curve 0 in the ^-plane and all points / within a given simple curve G' in the /-plane, let / (z, z) be a holomorphic function; then, if a be any point within C and a' any point within G', ^n+nJ (a, a') J (z - a)n+1 (z' — aTf where n and ri are any integers and the integral is taken positively round the two curves G and G'. If M be the greatest value of \f (z, z'} for points z and z within their respective regions when the curves G and G' are circles of radii r, r' and centres a, a', then dn+n'f(a, a') M ~3aW»' <w!/i!rv^5 F. 34 HOLOMORPHIC FUNCTIONS [22- M and if $(?>*) dn+n'f(a,a') dn+n'(j> (z, z') then da»da'« when z = a and z = a' in the derivative of <£ (z, z). 23. All the integrals of meromorphic functions that have been considered have been taken along complete curves : it is necessary to refer to integrals along curves which are lines only from one point to another. A single illustration will suffice at present. Consider the integral f -t-^-dz; the function /» is J H0 z — a supposed holomorphic in the given region, and z and z0 are any two points in that region. Let some curves joining z to z0 be drawn as in the figure (fig. 7). ~ , •* 2o is holomorphic over the whole area en- Fig> 7 z— a closed by z^zSz0: and therefore we have ^ = 0 when taken round the boundary of that area. Hence as in the earlier case we have z — a Jz0 z — a The point a lies within the area enclosed by z0yz^z0, and the function is holomorphic, except in the immediate vicinity of z = a ; hence r f ( v\ I - dz = 2Trif(a), J z — a the integral on the left-hand side being taken round Z0yzj3z0. Hence z — a Denoting ^-by g(z), the function g (z) has one pole a in the region £ "~ CL considered. The preceding results are connected only with the simplest form of meromorphic functions; other simple results can be derived by means of the other theorems proved in §§ 17—21. Those which have been obtained are sufficient however to shew that : The integral of a meromorphic function fg(z)dz from one point to another of the region of the function is not in general a uniform function. The value of the integral is not altered by any deformation of the path which does not meet or cross a pole of the function; but the value is altered when the path of integration is so 23.] GENERAL PROPOSITIONS IN INTEGRATION 35 deformed as to pass over one or more poles. Therefore it is necessary to specify the path of integration when the subject of integration is a mero- morphic function ; only partial deformations of the path of integration are possible without modifying the value of the integral. 24. The following additional propositions* are deduced from limiting cases of integration round complete curves. In the first, the curve becomes indefinitely small ; in the second, it becomes infinitely large. And in neither, are the properties of the functions to be integrated limited as in the pre ceding propositions, so that the results are of wider application. I. If f(z) be a function which, whatever be its character at a, has no infinities and no branch-points in the immediate vicinity of a, the value of ff(z)dz taken round a small circle with its centre at a tends towards zero when the circle diminishes in magnitude so as ultimately to be merely the point a, provided that, as z — a diminishes indefinitely, the limit of (z — a)f(z) tend uniformly to zero. Along the small circle, initially taken to be of radius r, let *-a-fl*i * dz so that = idO, z— a and therefore Sf(z) dz = i\ (z — a)f(z) d6. Jo Hence \ff(z)dz\ = I *" (z - a)f(z) d0 Jo <r\(z-a)f(z)\de Jo rzn < Md0 Jo where M' is the greatest value of M, the modulus of (z - a)f(z), for points on the circumference. Since (z - a)f(z) tends uniformly to the limit zero as | z -a diminishes indefinitely, \jf(z) dz\ is ultimately zero. Hence the integral itself jf(z)dz is zero, under the assigned conditions. Note. If the integral be extended over only part of the circumference of the circle, it is easy to see that, under the conditions of the proposition, the value offf(z)de still tends towards zero. COROLLARY. If (z-a)f(z) tend uniformly to a limit k as \z-a\ diminishes indefinitely, the value of ff(z)dz taken round a small circle centre a tends towards 27rik in the limit. * The form of the first two propositions, which is adopted here, is due to Jordan, Cours d' Analyse, t. ii, §§ 285, 286. 3—2 36 GENERAL PROPOSITIONS [24. Thus the value of [- dz j, taken round a very small circle centre «, where a is ~ d * /2V not the origin, is zero : the value of f - - - -, round the same circle is -. ( - \ . J (a — z) (a-M) Neither the theorem nor the corollary will apply to a function, such as sn —-^ which has the point a for an essential singularity: the value of (z-a)sn^— ^, as \z-a\ diminishes indefinitely, does not tend (§ 13) to a uniform limit. As a matter of fact the function sn — has an infinite number of poles in the immediate vicinity of a z- a as the limit z—a, is being reached. II. Whatever be the character of a function f (z} for infinitely large values ofz, the value ofjf(z) dz, taken round a circle with the origin for centre, tends towards zero as the circle becomes infinitely large, provided that, as \z\ increases indefinitely, the limit of zf(z) tend uniformly to zero. Along a circle, centre the origin and radius R, we have z =Eeei, so that dz .ja - = idd, z r-2ir and therefore // 0) dz = i zf(z) d6. Jo Hence I //(*)<&! = £* <T zf(z)\dS Jo rzn < Mde Jo < where M' is the greatest value of M, the modulus of zf(z)t for points on the circumference. When R increases indefinitely, the value of M' is zero on the hypothesis in the proposition; hence \$f(*)d*\ is ultimately zero. Therefore the value of ff(z) dz tends towards zero, under the assigned con ditions. Note. If the integral be extended along only a portion of the circumfer ence, the value of jf(z}dz still tends towards zero. COROLLARY. // zf(z) tend uniformly to a limit k as \z . increases indefinitely, the value of jf(z) dz, taken round a very large circle, centre the origin, tends towards %7rik. Thus the value of J(l -zn}~^dz round an infinitely large circle, centre the origin, is zero if n > 2, and is 2ir if » = 2. III. If all the infinities and the branch-points of a function lie in a finite region of the z-plane, then the value of jf(z) dz round any simple curve, which 24.] IN INTEGRATION 37 includes all those points, is zero, provided the value of zf(z\ as \z\ increases indefinitely, tends uniformly to zero. The simple curve can be deformed continuously into the infinite circle of the preceding proposition, without passing over any infinity or any branch- point ; hence, if we assume that the function exists all over the plane, the value of jf(z) dz is, by Cor. I. of § 19, equal to the value of the integral round the infinite circle, that is, by the preceding proposition, to zero. Another method of stating the proof of the theorem is to consider the corresponding simple curve on Neumann's sphere (§ 4). The surface of the sphere is divided into two portions by the curve*: in one portion lie all the singularities and the branch-points, and in the other portion there is no critical point whatever. Hence in this second portion the function is holo- morphic ; since the area is bounded by the curve we see that, on passing back to the plane, the excluded area is one over which the function is holomorphic. Hence, by § 19, the integral round the curve is equal to the integral round an infinite circle having its centre at the origin and is therefore zero, as before. COROLLARY. If, under the same circumstances, the value of zf(z}, as \z increases indefinitely, tend uniformly to k, then the value of $f(z)dz round the simple curve is Thus the value of I — — r along any simple curve which encloses the two points J (a2 - z2)* a and - a is 2ir ; the value of dz {(!-«") (!-*%•)}* round any simple curve enclosing the four points 1, -1, T, -7, is zero, k being a non- 1C K vanishing constant ; and the value of J(l — z2n)~*dz, taken round a circle, centre the origin and radius greater than unity, is zero when n is an integer greater than 1. /dz ~ ~ — 771 K*-«i) (*-««)(*-«•)}* round any circle, which has the origin for centre and includes the three distinct points €lt e2, e3, is not zero. The subject of integration has 2 = 00 for a branch-point, so that the condition in the proposition is not satisfied ; and the reason that the result is no longer valid is that the deformation into an infinite circle, as described in Cor. I. of § 19, is not possible because the infinite circle would meet the branch-point at infinity. 25. The further consideration of integrals of functions, that do not possess the character of uniformity over the whole area included by the curve of in tegration, will be deferred until Chap. ix. Some examples of the theorems proved in the present chapter will now be given. * The fact that a single path of integration is the boundary of two portions of the surface of the sphere, within which the function may have different characteristic properties, will be used hereafter (§ 104) to obtain a relation between the two integrals that arise according as the path is deformed within one portion or within the other. 38 EXAMPLES IN [25. Ex. 1. It is sufficient merely to mention the indefinite integrals (that is, integrals from an arbitrary point to a point z} of rational, integral, algebraical functions. After the preceding explanations it is evident that they follow the same laws as integrals of similar functions of real variables. /dz ,— ^ , taken round a simple curve. When n is 0, the value of the integral is zero if the curve do not include the point a, and it is Ziri if the curve include the point a. When n is a positive integer, the value of the integral is zero if the curve do not include the point a (by § 17), and the value of the integral is still zero if the curve do include the point a (by § 22, for the function f(z) of the text is 1 and all its derivatives are zero). Hence the value of the integral round any curve, which does not pass through a, is zero. We can now at once deduce, by § 20, the result that, if a holomorphic function be constant along any simple closed curve within its region, it is constant over the whole area within the curve. For let t be any point within the curve, z any point on it, and C the constant value of the function for all the points z ; then B' mn 2 — t the integral being taken round the curve, so that <&M-— t dz = C by the above result, since the point t lies within the curve. Ex. 3. Consider the integral \e~^dz. In any finite part of the plane, the function e~02 is holomorphic; therefore (§ 17) the integral round the boundary of a rectangle (fig. 8), bounded by the lines x= ±a, y = 0, y=b, is zero : and this boundary can be extended, provided the deformation remain in the region where the function is holo morphic. Now as a tends towards infinity, the modulus of e~z\ being e~x2 + y2, tends towards zero when y remains finite ; and therefore the preceding rectangle can be Fig. 8. extended towards infinity in the direction of the axis of x, the side b of the rectangle remaining unaltered. Along A' A, we have z=x : so that the value of the integral along the part A' A of the fa boundary is I e~x dx. J -a Along AB, we have z = a + iy, so that the value of the integral along the part AB f* is i I e~(a + iyrdy. Jo Along BB', we have z = x + ib, so that the value of the integral along the part BB' f'a is I e-(x + lVdx. J a Along B'A', we have z=-a + iy, so that the value of the integral along the part B'A1 is i (V(-«H J t, 25.] INTEGRATION 39 /•ft The second of these portions of the integral is e~a<i . » . I tP~***tefy, which is easily seen J o to be zero when the (real) quantity a is infinite. Similarly the fourth of these portions is zero. Hence as the complete integral is zero, we have, on passing to the limit, I e~^dx+\ e-^2ibx + b'2da;=0, J -<*> J oo whence e62 I e~ *-***&?=* I J -oo J -<* /oo e'3^ (cos 2bx—i sin and therefore, on equating real parts, we obtain the well-known result / J -Q This is only one of numerous examples* in which the theorems in the text can be applied to obtain the values of definite integrals with real limits and real variables. rzn-i Ex. 4. Consider the integral I - --- dz. where n is a real positive quantity less than J 1+z unity. The only infinities of the subject of integration are the origin and the point - 1 ; the branch-points are the origin and 2=00. Everywhere else in the plane the function behaves like a holomorphic function ; and, therefore, when we take any simple closed curve enclosing neither the origin nor the point — 1, the integral of the function round that curve is zero. We shall assume that the curve lies on the positive side of the axis of x and that it is made up of : — (i) a semicircle (73 (fig. 9), centre the origin and radius R which is made to increase indefinitely : Fig. 9. (ii) two semicircles, ct and c2, with their centres at 0 and — 1 respectively, and with radii r and /, which ultimately are made infinitesimally small : (iii) the diameter of (73 along the axis of x excepting those ultimately infinitesimal portions which are the diameters of cx and of c2. The subject of integration is uniform within the area thus enclosed although it is not uniform over the whole plane. We shall take that value of zn~l which has its argument equal to (n— 1) 6, where 6 is the argument of z. * See Briot and Bouquet, Theorie des fonctions elliptiques, (2nd ed.), pp. 141 et sqq., from which examples 3 and 4 are taken. 40 EXAMPLES IN [25. The integral round the boundary is made up of four parts. 0H — 1 (a) The integral round (73. The value of z . , as z \ increases indefinitely, tends uniformly to the limit zero ; hence, as the radius of the semicircle is increased indefinitely, the integral round (73 vanishes (§ 24, n., Note). ^n— 1 (b) The integral round cv The value of z . , as | z \ diminishes indefinitely, 1 -\-z tends uniformly to the limit zero ; hence as the radius of the semicircle is diminished indefinitely, the integral round cv vanishes (§ 24, I., Note}. zn-l (c) The integral round c2. The value of (1 + 2) , as |1+2| diminishes indefinitely A ~r z for points in the area, tends uniformly to the limit (— I)""1, i.e., to the limit g(M~1)'™. Hence this part of the integral is being taken in the direction indicated by the arrow round c2) the infinitesimal semicircle. Evidently -- =id6 and the limits are TT to 0, so that this part of the whole integral is idd (d) The integral along the axis of x. The parts at — 1 and at 0 which form the diameters of the small semicircles are to be omitted ; so that the value is -l+r' J r This is what Cauchy calls the principal value* of the integral /"°° /yH ~ 1 / •* 7 I dx. Since the whole integral is zero, we have ineniri+ I Y — dx = 0. Let P = I ^ — dx, P' = I dx, and 0— I dx, J o 1-a? principal values being taken in each case. Then, taking account of the arguments, we have Since iwenvi + P + 1* = 0, we have P - eH7riQ = - inenni, * Williamson's Integral Calculus, % 104. 25.] INTEGRATION 41 so that P— Q cos nn — ir sin nn, Q sin ntr = TT cos nir. Hence I ; dx—P = ir cosec TOTT, jo 1+a? dx—Q = ir cot %TT. . 5. In the same way it may be proved that . where n is an integer, a is positive and o> is e*2" . Jik 6. By considering the integral Je-2^™-1^ round the contour of the sector of a circle of radius r, bounded by the radii 0=0, 6=a, where a is less than |TT and n is positive, it may be proved that „»— 1 „-; {r'; on proceeding to the limit when r is made infinite. (Briot and Bouquet.) Ex. 7. Consider the integral I ~~^, where n is an integer. The subject of integration is meromorphic ; it has for its poles (each of which is simple) the n points o>r for r=0, 1, ..., n-l, where a is a primitive nth root of unity ; and it has no other infinities and no branch -points. Moreover the value of — — -, as \z\ increases indefinitely, tends uniformly to the limit zero ; hence (§ 24, in.) the value of the integral, taken round a circle centre the origin and radius > 1, is zero. This result can be derived by means of Corollary II. in § 19. Surround each of the poles with an infinitesimal circle having the pole for centre ; then the integral round the circle of radius > 1 is equal to the sum of the values of the integral round the infinitesimal circles. The value round the circle having «r for its centre is, by § 20, 2rri( limit of " , when z = u>r} \ z - L J Hence the integral round the large circle 2 n r=n = 0. Ex. 8. Hitherto, in all the examples considered, the poles that have occurred have been simple : but the results proved in § 21 enable us to obtain the integrals of functions which have multiple poles within an area. As an example, consider the integral / (1+g2)n + i round any curve which includes the point i but not the point - i, these points being the two poles of the subject of integration, each of multiplicity n + l. 42 EXAMPLES IN INTEGRATION [25. We have seen that /" (a) = j^ J ^_a^n + i *» where /(«) is holomorphic throughout the region bounded by the curve round which the integral is taken. In the present case a is i, and f(z) = . «.n ^\ ', s° that 2» ! (-l)n u. <• /*n - 2- and therefore /" (*J = ^j (2i)»*-n "~ ~ wT Hence we have "***™' In the case of the integral of a function round a simple curve which contains several of its poles we first (§ 20) resolve the integral into the sum of the integrals round simple curves each containing only one of the points, and then determine each of the latter integrals as above. Another method that is sometimes possible makes use of the expression of the uniform function in partial fractions. After Ex. 2, we need retain only those fractions which are of the form — : the integral of such a fraction is ZniA, and the value of the whole integral z-a is therefore tor&A. It is thus sufficient to obtain the coefficients of the inverse first powers which arise when the function is expressed in partial fractions corresponding to each pole. Such a coefficient A, the coefficient of -j in the expansion of the function, is called by Z (Jj Cauchy the residue of the function relative to the point. For example, so that the residues relative to the points -1, -o>, -to2 are f, £«, |«2 respectively. Hence if we take a semicircle, of radius > 1 and centre the origin with its diameter along the axis of y, so as to lie on the positive side of the axis of y, the area between the semi-circumference and the diameter includes the two points -« and -«2 ; and therefore the value of dz taken along the semi-circumference and the diameter, is &*&»+!•?); i.e., the value is - *ni. CHAPTER III. EXPANSION OF FUNCTIONS IN SERIES OF POWERS. 26. WE are now in a position to obtain the two fundamental theorems relating to the expansion of functions in series of powers of the variable : they are due to Cauchy and Laurent respectively. Cauchy 's theorem is as follows*: — When a function is holomorphic over the area of a circle of centre a, it can be expanded as a series of positive integral powers of z-a converging for all points within the circle. Let z be any point within the circle; describe a concentric circle of radius r such that \z-a\ = p <r<R, ^ ^i. where R is the radius of the given circle. If t denote a current point on the circumference of the new circle, we have dt t — a z — a t — a Fif?. 10. the integral extending along the whole circumference of radius r. Now z-a t-a z -an+l z — a — a t—a so that, by § 14 (III.), we have J_ f 27ri] f(t) t-z\t-a dt. * Exercices d' 'Analyse et de Physique Mathe'matiqne, t. ii, pp. 50 et seq. ; the memoir was first made public at Turin in 1832. 44 CAUCHY'S THEOREM ON THE [26. Now /(«) is holomorphic over the whole area of the circle ; hence, if t be not actually on the boundary of the region (§§ 21, 22), a condition secured by the hypothesis r < R, we have and therefore (z-a)n (z-a)n+l " Let the last term be denoted by L. Since z — a =p and \t-a\ = r, it is at once evident that \t-z\^r-p. Let M be the greatest value of |/(0| for points along the circle of radius r ; then M must be finite, owing to the initial hypothesis relating tof(z). Taking f — n — TP6i v W/ ~~ I C* so that dt = i(t- a) d6, P«+> t*m de we have \L\ = -f i ^P Jlf rn (i — p) Jffl-:C \r) Now r was chosen to be greater than p ; hence as n becomes infinitely large, we have W infinitesimally small. Also If (1 — ?l is finite. \r/ V f/ Hence as ?i increases indefinitely, the limit of |i|, necessarily not negative, is infinitesimally small and therefore, in the same case, L tends towards zero. It thus appears, exactly as in § 15 (V.), that, when n is made to increase without limit, the difference between the quantity f(z) and the first n + 1 terms of the series is ultimately zero ; hence the series is a converging series having f(z) as the limit of the sum, so that which proves the proposition under the assigned conditions. It is the form of Taylor's expansion for complex variables. Note. The series on the right-hand side is frequently denoted by P(z — a), where P is a general symbol for a converging series of positive integral powers of z — a: it is also sometimes* denoted by P(z\a). Con- * Weierstrass, Abh. am der Functionenlehre, p. 1. 26.] EXPANSION OF A FUNCTION 45 formably with this notation, a series of negative integral powers of z — a would be denoted by P I - — ) ; a series of negative integral powers of z \z — a/ either by P (-) or by P(^|oo), the latter implying a series proceeding in \zj positive integral powers of a quantity which vanishes when z is infinite, i.e., in positive integral powers of — . Z If, however, the circle can be made of infinitely great radius so that the function f(z) is holomorphic over the finite part of the plane, the equivalent series is denoted by G(z — a) and it converges over the whole plane. Conformably with this notation, a series of negative integral powers of z - a which converges over the whole plane is denoted by G I - - j . 27. The following remarks on the proof and on inferences from it should be noticed. (i) In order that — - - may be expanded in the required form, the t — z point z must be taken actually within the area of the circle of radius R ; and therefore the convergence of the series P (z — a) is not established for points on the circumference. (ii) The coefficients of the powers of z — a in the series are the values of the function and its derivatives at the centre of the circle ; and the character of the derivatives is sufficiently ensured (§ 21) by the holomorphic character of the function for all points within the region. It therefore follows that, if a function be holomorphic within a region bounded by a circle of centre a, its expansion in a series of ascending powers of z — a converging for all points within the circle depends only upon the values of the function and its derivatives at the centre. But instead of having the values of the function and of all its derivatives at the centre of the circle, it will suffice to have the values of the holomorphic function itself over any small region at a or along any small line through a, the region or the line not being infinitesimal. The values of the derivatives at a can be found in either case ; for /' (b) is the limit of {f(b + 86) —f(b)}/8b, so that the value of the first derivative can be found for any point in the region or on the line, as the case may be ; and so for all the derivatives in succession. (iii) The form of Maclaurin's series for complex variables is at once derivable by supposing the centre of the circle at the origin. We then infer that, if a function be kolomorphic over a circle, centre tJie origin, it can be 46 DARBOUX'S EXPRESSION [27. represented in the form of a series of ascending, positive, integral powers of the variable given by where the coefficients of the various powers of z are the values of the derivatives of f(z) at the origin, and the series converges for all points within the circle. Thus, the function ez is holomorphic over the finite part of the plane ; therefore its expansion is of the form G (z). The function log (1 4- z) has a singularity at — 1 ; hence within a circle, centre the origin and radius unity, it can be expanded in the form of an ascending series of positive integral powers of z, it being convenient to choose that one of the values of the function which is zero at the origin. Again, tan"1.?2 has singularities at the four points z4 = — I, which all lie on the circumference; choosing the value at the origin which is zero there, we have a similar expansion in a series, con verging for points within the circle. Similarly for the function (1 +z)n, which has — 1 for a singularity. (iv) Darboux's method* of derivation of the expansion of f (z) in positive powers of z — a depends upon the expression, obtained in § 15 (IV.), for the value of an integral. When applied to the general term 1 Uz-a\n+i s,.. ,, f(t)dt, = L say, it gives L = \r fe^J /(f), where £ is some point on the circumference of the circle of radius r, and X is 2 ~ fl a complex quantity of modulus not greater than unity. The modulus of ^ b ~~ a is less than a quantity which is less than unity ; the terms of the series of moduli are therefore less than the terms of a converging geometric progres sion, so that they form a converging series; the limit of \L\, and therefore of L, can, with indefinite increase of n, be made zero and Taylor's expansion can be derived as before. 00 Ex. 1. Prove that the arithmetic mean of all values of z~ n 2 avzv, for points lying along v = 0 a circle |z| = r entirely contained in the region of continuity, is an. (Rouche, Gutzmer.) Prove also that the arithmetic mean of the squares of the moduli of all values of 00 2 avzv, for points lying along a circle z\ = r entirely contained in the region of continuity, x = 0 is equal to the sum of the squares of the moduli of the terms of the series for a point on the circle. (Gutzmer.) 00 Ex. 2. Prove that the function 2 anzn*, M = 0 is finite and continuous, as well as all its derivatives, within and on the boundary of the circle |0| = 1, provided a < 1. (Fredholm.) * Liouville, 3dmc Ser., t. ii, (1876), pp. 291—312. 28.] LAURENT'S EXPANSION OF A FUNCTION 47 28. Laurent's theorem is as follows*: — A function, which is holomorphic in a part of the plane bounded by two concentric circles with centre a and finite radii, can be expanded in the form of a double series of integral powers, positive and negative, of z — a, the series converging uniformly and unconditionally in the part of the plane between the circles. Let z be any point within the region bounded by the two circles of radii R and R; describe two concentric circles of radii r arid r' such that R>r> z-a >r'> R. Denoting by t and by s current points on the circumference of the outer and of the inner circles respectively, and considering the space which lies between them and includes the point z, we have, by § 20, /w-oL : — Z ZTTlJs — 2"~ Fig. 11. a negative sign being prefixed to the second integral because the direction indicated in the figure is the negative direction for the description of the inner circle regarded as a portion of the boundary. Now we have fz — a" t — a _ z — t ' — a Iz — a\* — a \t-aj z- a. + I . — - + — a. 1 - z — a t — a this expansion being adopted with a view to an infinite converging series, z — a because t — a is less than unity for all points t; and hence, by § 15, _ n\n+l dt. — z \t — a/ Now each of the integrals, which are the respective coefficients of powers of z — a, is finite, because the subject of integration is everywhere finite along the circle of finite radius, by § 15 (IV.). Let the value of ^r* % '•'-'••- be 2iriur : the quantity ur is not necessarily equal to /'' (a) -r- r I, because no * Comptes Rendus, t. xvii, (1843), p. 939. 48 LAURENT'S EXPANSION OF [28. knowledge of the function or of its derivatives is given for a point within the innermost circle of radius R'. Thus _L f/2) dt = u0 + (z - a) u1 + (z- a)2 w2+ +(z- a)nun 2w» J t — z 1 [f (t) (z — a\n+1 -, - z \t — a The modulus of the last term is less than M where p is z-a and If is the greatest value of \f(t)\ for points along the circle. Because p < r, this quantity diminishes to zero with indefinite in crease of n ; and therefore the modulus of the expression v % becomes indefinitely small with increase of n. The quantity itself therefore vanishes in the same limiting circumstance ; and hence 1 . [fl&dt = u0 + (z-<i)u1 + ...... +(z-a)mum+ ...... , 2-7TI J t — Z so that the first of the integrals is equal to a series of positive powers. This series converges uniformly and unconditionally within the outer circle, for the modulus of the (m + l)th term is less than which is the (m + l)th term of a converging series*. As in § 27, the equivalence of the integral and the series can be affirmed only for points which lie within the outermost circle of radius R. Again, we have fs - a\n+1 z-a _ s-a fs - a\n (z-a) s-z z-a \z-a z — a this expansion being adopted with a view to an infinite converging series, because s — a z — a is less than unity. Hence 1 [/s-a\ . If - - 2?rt J \z-aj -n+1f(s) J-~- , -ds. z — s Chrystal, ii, 124. 28.] A FUNCTION IN SERIES 49 The modulus of the last term is less than M' P where M' is the greatest value of \f(s)\ for points along the circle of radius r'. With indefinite increase of n, this modulus is ultimately zero ; and thus, by an argument similar to the one which was applied to the former integral, we have .. .. - .. ZTTI J s — z z — a (z — a)2 (z — a)m where vm denotes the integral f(s — a)m~lf (s) ds taken round the circle. As in the former case, the series is one which converges uniformly and unconditionally; and the equivalence of the integral and the series is valid for points z that lie without the innermost circle of radius R'. The coefficients of the various negative powers of z — a are of the form 1 f /(*) d( 1 ^ tori] __ 1_ (s-a)' (s - a)m a form that suggests values of the derivatives of f (s) at the point given by - = 0, that is, at infinity. But the outermost circle is of finite radius ; s-a and no knowledge of the function at infinity, lying without the circle, is given, so that the coefficients of the negative powers may not be assumed to be the values of the derivatives at infinity, just as, in the former case, the coefficients ur could not be assumed to be the values of the derivatives at the common centres of the circles. Combining the expressions obtained for the two integrals, we have f(z) = u0 + (z — a) u-i + (z — a)2 w2 + ... + (z- a)-1 Vl + (z- a)~2 va+ .... Both parts of the double series converge uniformly and unconditionally for all points in the region between the two circles, though not necessarily for points on the boundary of the region. The whole series therefore converges for all those points : and we infer the theorem as enunciated. Conformably with the notation (§ 26, note) adopted to represent Taylor's expansion, a function f(z) of the character required by Laurent's Theorem can be represented in the form the series P1 converging within the outer circle and the series P2 converging without the inner circle ; their sum converges for the ring-space between the circles. F. 4 50 LAURENT'S THEOREM [29. 29. The coefficient u0 in the foregoing expansion is -1- f £9 dt torijt-a ' the integral being taken round the circle of radius r. We have dt =ide t — a for points on the circle ; and therefore d0 so that \u0\<!deMt<M', J ZTT M' being the greatest value of Mt, the modulus of f(t), for points along the circle. If M be the greatest value of \f(z}\ for any point in the whole region in which f(z) is defined, so that M'^.M, then we have «o 1 < M, that is, the modulus of the term independent of z — a in the expansion of f(z) by Laurent's Theorem is less than the greatest value of \f(z) \ at points in the region in which it is defined. Again, (z-a)-mf(z) is a double series in positive and negative powers of z-a, the term independent of z -a being um; hence, by what has just been proved, um \ is less than p~m M, where p is z - a . But the coefficient um does not involve z, and we can therefore choose a limit for any point z. The lowest limit will evidently be given by taking z on the outer circle of radius R, so that um < MR~m. Similarly for the coefficients vm ; and therefore we have the result : — If f(z) be expanded as by Laurent's Theorem in the form OO 00 u0+ 2 (z-a)mum+ 2 (z-aY^Vm, m = l m=l then \um <MR~m, \vm <MR'm, where M is the greatest value of \f(z) at points within the region in which f(z) is defined, and R and R' are the radii of the outer and the inner circles respectively. 30. The following proposition is practically a corollary from Laurent's Theorem : — When a function is holomorphic over all the plane which lies outside a circle of centre a, it can be expanded in the form of a series of negative integral powers of z — a, the series converging uniformly and unconditionally everywhere in that part of the plane. It can be deduced as the limiting case of Laurent's Theorem when the 30.] EXPANSION IN NEGATIVE POWERS 51 radius of the outer circle is made infinite. We then take r infinitely large, and substitute for t by the relation t — a = reei, so that the first integral in the expression (a), p. 47, for/(^) is 1 f2" d0 t — a Since the function is holomorphic over the whole of the plane which lies outside the assigned circle, f(t} cannot be infinite at the circle of radius r when that radius increases indefinitely. If it tend towards a (finite) limit k, which must be uniform owing to the hypothesis as to the functional character of f(z\ then, since the limit of (t — z)/(t — a) is unity, the preceding integral is equal to k. The second integral in the same expression (a), p. 47, for f(z) is un altered by the conditions of the present proposition ; hence we have f(z) = k + (z- a)~l vl + (z- a)-2Vz + ..., the series converging uniformly and unconditionally without the circle, though it does not necessarily converge on the circumference. The series can be represented in the form 1 \z — a/ conformably with the notation of § 26. Of the three theorems in expansion which have been obtained, Cauchy's is the most definite, because the coefficients of the powers are explicitly obtained as values of the function and of its derivatives at an assigned point. In Laurent's theorem, the coefficients are not evaluated into simple expres sions ; and in the corollary frofti Laurent's theorem the coefficients are, as is easily proved, the values of the function and of its derivatives for infinite values of the variable. The essentially important feature of all the theorems is the expansibility of the function in series under assigned conditions. 31. It was proved (§21) that, when a function is holomorphic in any region of the plane bounded by a simple curve, it has an unlimited number of successive derivatives each of which is holomorphic in the region. Hence, by the preceding propositions, each such derivative can be expanded in converging series of integral powers, the series themselves being deducible by differentiation from the series which represents the function in the region. In particular, when the region is a finite circle of centre a, within which f(z) and consequently all the derivatives off(z) are expansible in converging series of positive integral powers of z — a, the coefficients of the various powers of z — a are — save as to numerical factors — the values of the 4—2 52 DEFINITION OF DOMAIN [31. derivatives at the centre of the circle. Hence it appears that, when a function is holomorphic over the area of a given circle, the values of the function and all its derivatives at any point z within the circle depend only upon the variable of the point and upon the values of the function and its derivatives at the centre. 32. Some of the classes of points in a plane that usually arise in connection with uniform functions may now be considered. (i) A point a in the plane may be such that a function of the variable has a determinate finite value there, always independent of the path by which the variable reaches a ; the point a, is called an ordinary point* of the function. The function, supposed continuous in the vicinity of a, is con tinuous at a : and it is said to behave regularly in the vicinity of an ordinary point. Let such an ordinary point a be at a distance d, not infinitesimal, from the nearest of the singular points (if any) of the function ; and let a circle of centre a and radius just less than d be drawn. The part of the z-plane lying within this circle is calledf the domain of a ; and the function, holomorphic within this circle, is said to behave regularly (or to be regular) in the domain of a. From the preceding section, we infer that a function and its derivatives can be expanded in a converging series of positive integral powers of z — a for all points z in the domain of a, an ordinary point of the function : and the coefficients in the series are the values of the function and its derivatives at a. The property possessed by the series — that it contains only positive integral powers of z - a— at once gives a test that is both necessary and sufficient to determine whether a point is an ordinary point. If the point a be ordinary, the limit of (z - a) f (z} necessarily is zero when z becomes equal to a. This necessary condition is also sufficient to ensure that the point is an ordinary point of the function / (z), supposed to be uniform ; for, since f(z) is holomorphic, the function (z-a)f(z) is also holomorphic and can be expanded in a series M0 -f wa (z — d) + w2 (? — a)2 + • • -, converging in the domain of a. The quantity u0 is zero, being the value of (z-a)f(z) at a and this vanishes by hypothesis; hence (z-a)f (z) = (z — a) {MI + u2(z -a) +...}, shewing that / (z) is expressible as a series of positive integral powers of z— a converging within the domain of a, or, in other words, that/(*) certainly has a for an ordinary point in consequence of the condition being satisfied. * Sometimes a regular point. t The German title is Umgebung, the French is domaine. 32.] ESSENTIAL SINGULARITY 53 (ii) A point a in the plane may be such that a function / (z) of the variable has a determinate infinite value there, always independent of the path by which the variable reaches a, the function behaving regularly for points in the vicinity of a ; then ^—\ nas a determinate zero value there, so / (?) that a is an ordinary point of --r-r . The point a is called a pole (§12) or an accidental singularity* of the function. A test, necessary and sufficient to settle whether a point is an accidental singularity of a function will subsequently (§ 42) be given. (iii) A point a in the plane may be such that y (2) has not a determinate value there, either finite or infinite, though the function is regular for all points in the vicinity of a that are not at merely infinitesimal distances. i 1 Thus the origin is of this nature for the functions ez, sn - . Z Such a point is called-f* an essential singularity of the function. No hypothesis is postulated as to the character of the function for points at infinitesimal distances from the essential singularity, while the relation of the singularity to the function naturally depends upon this character at points near it. There may thus be various kinds of essential singularities all included under the foregoing definition ; their classification is effected through the consideration of the character of the function at points in their immediate vicinity. (See § 88.) One sufficient test of discrimination between an accidental singularity and an essential singularity is furnished by the determinateness of the value at the point. If the reciprocal of the function have the point for an ordinary point, the point is an accidental singularity — it is, indeed, a zero for the reciprocal. But when the point is an essential singularity, the value of the reciprocal of the function is not determinate there ; and then the reciprocal, as well as the function, has the point for an essential singularity. 33. It may be remarked at once that there must be at least one infinite value among the values which a function can assume at an essential singularity. For if/ (z) cannot be infinite at a, then the limit of (z — a)f (z) is zero when z = a, no matter what the non-infinite values of f (z) may be, that is, the limit is a determinate zero. The function (z — a)f(z) is regular in the vicinity of a : hence by the foregoing test for an ordinary point, the point a is ordinary and the value of the uniform function f(z) is * Weierstrass, Abh. aus der Functionenlehre, p. 2, to whom the name is due, calls it ausser- wesentliche singuldre Stelle ; the term non-essential is suggested by Mr Cathcart, Harnack, p. 148. t Weierstrass, I.e., calls it wesentliche singulare Stelle. 54 CONTINUATIONS OF A FUNCTION [33. determinate, contrary to hypothesis. Hence the function must have at least one infinite value at an essential singularity. Further, a uniform function must be capable of assuming any value C at an essential singularity. For an essential singularity of / (z) is also an essential singularity of / (z) — G and therefore also of .. \_n • The last function must have at least one infinite value among the values that it can assume at the point ; and, for this infinite value, we have / (z) — C at the point, so that/(f) assumes the assigned value C at the essential singularity*. 34. Let f(z) denote the function represented by a series of powers Pj (z — a), the circle of convergence of which is the domain of the ordinary point a of the function. The region over which the function / (z) is holo- morphic may extend beyond the domain of a, although the circumference bounding that domain is the greatest of centre a that can be drawn within the region. The region evidently cannot extend beyond the domain of a in all directions. Take an ordinary point b in the domain of a. The value at b of the function /(V) is given by the series Pj (b — a), and the values at b of all its derivatives are given by the derived series. All these series converge within the domain of a and they are therefore finite at b ; and their expressions involve the values at a of the function and its derivatives. Let the domain of b be formed. The domain of b may be included in that of a, and then its bounding circle will touch the bounding circle of the domain of a internally. If the domain of b be not entirely included in that of a, part of it will lie outside the domain of a ; but it cannot include the whole of the domain of a unless its bounding circumference touch that of the domain of a externally, for otherwise it would extend beyond a in all directions, a result inconsistent with the construction of the domain of a. Hence there must be points excluded from the domain of a which are also excluded from the domain of b. For all points z in the domain of b, the function can be represented by a series, say P2 (2 — b), the coefficients of which are the values at b of the function and its derivatives. Since these values are partially dependent upon the corresponding values at a, the series representing the function may be denoted by P2 (z — b, a). At a point z in the domain of b lying also in the domain of a, the two series Pl (z — a) and P2 (z — b, a) must furnish the same value for the function / (V) ; and therefore no new value is derived from the new series P2 * Weierstrass, I.e., pp. 50—52; Durege, Elemente der Theorie der Funktionen, p. 119; Holder, Math. Ann., t. xx, (1882), pp. 138 — 143 ; Picard, " Memoire sur les fonctions entieres," Annahs de VEcole Norm. Sup., 2me Ser., t. ix, (1880), pp. 145 — 166, which, in this regard, should be consulted in connection with the developments in Chapter V. See also § 62. 34.] OVER ITS REGION OF CONTINUITY 55 which cannot be derived from the old series Pj. For all such points the new series is of no advantage ; and hence, if the domain of b be included in that of a, the construction of the series P2 (z — b, a) is superfluous. Hence in choosing the ordinary point b in the domain of a we choose a point, if possible, that will not have its domain included in that of a. At a point z in the domain of b, which does not lie in the domain of a, the series P2 (z — b, a) gives a value for f(z) which cannot be given by Pl (z — a). The new series P2 then gives an additional representation of the function ; it is called* a continuation of the series which represents the function in the domain of a. The derivatives of P2 give the values of f(z) for points in the domain of b. It thus appears that, if the whole of the domain of b be not included in that of a, the function can, by the series which is valid over the whole of the new domain, be continued into that part of the new domain excluded from the domain of a. Now take a point c within the region occupied by the combined domains of a and b ; and construct the domain of c. In the new domain, the function can be represented by a new series, say P3(z — c), or, since the coefficients (being the values at c of the function and of its derivatives) involve the values at a and possibly also the values at b of the function and of its derivatives, the series representing the function may be denoted by Pz(z — c, a, b). Unless the domain of c include points, which are not included in the combined domains of a and b, the series P3 does not give a value of the function which cannot be given by Pj or P2: we therefore choose c, if possible, so that its domain will include points not included in the earlier domains. At such points z in the domain of c as are excluded from the combined domains of a and 6, the series P3 (z — c, a, b) gives a value for f(z) which cannot be derived from P1 or P2 ; and thus the new series is a continuation of the earlier series. Proceeding in this manner by taking successive points and constructing their domains, we can reach all parts of the plane connected with one another where the function preserves its holomorphic character; their combined aggregate is called -f the region of continuity of the function. With each domain, constructed so as to include some portion of the region of continuity not included in the earlier domains, a series is associated, which is a continuation of the earlier series and, as such, gives a value of the function not deducible from those earlier series ; and all the associated series are ultimately derived from the first. * Biermann, Theorie der analytischen Functional, p. 170, which may be consulted in connection with the whole of § 34; the German word is Fortsetzung. t Weierstrass, I.e., p. 1. 56 DEFINITION OF ANALYTIC FUNCTION [34. Each of the continuations is called an Element of the function. The aggregate of all the distinct elements is called a monogenic analytic function : it is evidently the complete analytical expression of the function in its region of continuity. Let z be any point in the region of continuity, not necessarily in the circle of convergence of the initial element of the function; a value of the function at z can be obtained through the continuations of that initial element. In the formation of each new domain (and therefore of each new element) a certain amount of arbitrary choice is possible ; and there may, moreover, be different sets of domains which, taken together in a set, each lead to z from the initial point. When the analytic function is uniform, as before defined (§ 12), the same value at z for the function is obtained, whatever be the set of domains. If there be two sets of elements, differently obtained, which give at z different values for the function, then the ana lytic function is multiform, as before defined (§ 12) ; but not every change in a set of elements leads to a change in the value at z of a multiform function, and the analytic function is uniform within such a region of the plane as admits only equivalent changes of elements. The whole process is reversible when the function is uniform. We can pass back from any point to any earlier point by the use, if necessary, of intermediate points. Thus, if the point a in the foregoing explanation be not included in the domain of b (there supposed to contribute a continu ation of the first series), an intermediate point on a line, drawn in the region of continuity so as to join a and b, would be taken ; and so on, until a domain is formed which does include a. The continuation, associated with this domain, must give at a the proper value for the function and its derivatives, and therefore for the domain of a the original series Pl(z — a) will be obtained, that is, Pj (z — a) can be deduced from P2 (z — b, a) the series in the domain of b. This result is general, so that any one of the continuations of a uniform function, represented by a power-series, can be derived from any other; and therefore the expression of such a function in its region of continuity is potentially given by one element, for all the distinct elements can be derived from any one element. 35. It has been assumed that the property, characteristic of some of the functions adduced as examples, of possessing either accidental or essential singularities, is characteristic of all functions ; it will be proved (§ 40) to hold for every uniform function which is not a mere constant. The singularities limit the region of continuity ; for each of the separate domains is, from its construction, limited by the nearest singularity, and the combined aggregate of the domains constitutes the region of continuity when 35.] SCHWARZ S CONTINUATION 57 they form a continuous space*. Hence the complete boundary of the region of continuity is the aggregate of the singularities of the function-}-. It may happen that a function has no singularity except at infinity ; the region of continuity then extends over the whole finite part of the plane but it does not include the point at infinity. It follows from the foregoing explanations that, in order to know a uniform analytic function, it is necessary to know some element of the function, which has been shewn to be potentially sufficient for the derivation of the full expression of the function and for the construction of its region of continuity. 36. The method of continuation of a function, which has just been described, is quite general ; there is one particular continuation, which is important in investigations on conformal representations. It is contained in the following proposition, due to SchwarzJ : — If an analytic function w of z be defined only for a region 8' in the positive half of the z-plane and if continuous real values of w correspond to continuous real values of z, then w can be continued across the axis of real quantities. Consider a region 8", symmetrical with S' relative to the axis of real quantities (fig. 12). Then a function is defined for the region S" by associating a value w0, the conjugate of w, with z0, the conjugate of z. Let the two regions be combined along the portion of the axis of ac which is their common boundary ; they then form a single region S' + S". Consider the integrals Fig. 12. 1 [ w j A ! [ wo o — • I — i-dz and ^ — -. / — fcp/fjr-f 2w»./,r«t- taken round the boundaries of 8' and of 8" respectively. Since w is * Cases occur in which the region of continuity of a function is composed of isolated spaces, each continuous in itself, but not continuous into one another. The consideration of such cases will be dealt with briefly hereafter, and they are assumed excluded for the present : meanwhile, it is sufficient to note that each continuous space could be derived from an element belonging to some domain of that space and that a new element would be needed for a new space. t See Weierstrass, I.e., pp. 1—3 ; Mittag-Leffler, " Sur la representation analytique des fonctions monogenes uniformes d'une variable independante," Acta Math., t. iv, (1884), pp. 1 et seq., especially pp. 1 — 8. £ Crelle, t. Ixx, (1869), pp. 106, 107, and Ges. Math. Abh., t. ii, pp. 66—68. See also Darboux, Theorie generate des surfaces, t. i, § 130. 58 SCHWARZ'S CONTINUATION [36. continuous over the whole area of 8' as well as along its boundary and likewise w0 relative to 8", it follows that, if the point f be in 8', the value of the first integral is w (f ) and that of the second is zero ; while, if £ lie in 8", the value of the first integral is zero and that of the second is w0 (£). Hence the sum of the two integrals represents a unique function of a point in either 8' or 8". But the value of the first integral is M wdz J^ [B w Q) dap I ("' ~~ (f • C\ ' I V> J ji 2— £ ZTriJ A x — L, the first being taken along the curve EC. A and the second along the axis AxB ; and the value of the second integral is 1 CAw0(x)dx 1_ f * W0dz0 2-Tri J B x — £ ZTTI J A *o — £ ' the first being taken along the axis Ex A and the second along the curve ADB. But w0 (ac) = w (x), because conjugate values w and w0 correspond to conjugate values of the argument by definition of W0 and because w (and therefore also w0) is real and continuous when the argument is real and continuous. Hence when the sum of the four integrals is taken, the two integrals corresponding to the two descriptions of the axis of x cancel and we have as the sum wdz 1 A and this sum represents a unique function of a point in 8' + 8". These two integrals, taken together, are _L [w'dz 2Tn]z-t' taken round the whole contour of 8' + 8", where w' is equal to w (f) in the positive half of the plane and to w0 (^) in the negative half. For all points £ in the whole region 8' + 8", this integral represents a single uniform, finite, continuous function of f; its value is w (£) in the positive half of the plane and is w0 (f) in the negative half; and therefore w0 (£) is the continuation into the negative half of the plane of the function, which is defined by w (£) for the positive half. For a point c on the axis of x, we have w (z) -w(c) = A(z-c) + B(z-cy>+C(z-cY + ...; and all the coefficients A, B, C,... are real. If, in addition, w be such a function of z that the inverse functional relation makes z a uniform analytic function of w, it is easy to see that A must not vanish, so that the functional relation may be expressed in the form w(z)—w (c) = (z-c}P(z- c), where P (z — c) does not vanish when z = c. CHAPTER IV. GENERAL PROPERTIES OF UNIFORM FUNCTIONS, PARTICULARLY OF THOSE WITHOUT ESSENTIAL SINGULARITIES. 37. IN the derivation of the general properties of functions, which will be deduced in the present and the next three chapters from the results already obtained, it is to be supposed, in the absence of any express statement to other effect, that the functions are uniform, monogenic and, except at either accidental or essential singularities, continuous*. THEOREM I. A function, which is constant throughout any region of the plane not infinitesimal in area, or which is constant along any line not infini tesimal in length, is constant throughout its region of continuity. For the first part of the theorem, we take any point a in the region of the plane where the function is constant, and we draw a circle of centre a and of any radius, provided only that the circle remains within the region of continuity of the function. At any point z within this circle we have /<*) =/(a) + (z - a)f (a) + (-, ~^ f" (a) + . . , a converging series the coefficients of which are the values of the function and its derivatives at a. But /X«) = Limit of ^±MZ/^), :. V, ' :• which is zero because f(a + Ba) is the same constant as f(a) : so that the first derivative is zero at a. Similarly, all the derivatives can be shewn to be zero at a ; hence the above series after its first term is evanescent, and we have /(*)-/<«), that is, the function preserves its constant value throughout its region of continuity. The second result follows in the same way, -when once the derivatives are proved zero. Since the function is monogenic, the value of the first and * It will be assumed, as in § 35 (note, p. 57), that the region of continuity consists of a single space ; functions, with regions of continuity consisting of a number of separated spaces, will be discussed in Chap. VII. 60 ZEROS OF A [37. of each of the successive derivatives will be obtained, if we make the differential element of the independent variable vanish along the line. Now, if a be a point on the line and a + 8a a consecutive point, we have f(a + So) = f(a) ; hence /' (a) is zero. Similarly the first derivative at any other point on the line is zero. Therefore we have /' (a + So) =f (a), for each has just been proved to be zero : hence /" (a) is zero ; and similarly the value of the second derivative at any other point on the line is zero. So on for all the derivatives : the value of each of them at a is zero. Using the same expansion as before and inserting again the zero values of all the derivatives at a, we find that /(*)=/(«), so that under the assigned condition the function preserves its constant value throughout its region of continuity. It should be noted that, if in the first case the area be so infinitesimally small and in the second the line be so infinitesimally short that consecutive points cannot be taken, then the values at a of the derivatives cannot be proved to be zero and the theorem cannot then be inferred. COROLLARY I. If two functions have the same value over any area of their common region of continuity which is not infinitesimally small or along any line in that region which is not infinitesimally short, then they have the same values at all points in their common region of continuity. This is at once evident : for their difference is zero over that area or along that line and therefore, by the preceding theorem, their difference has a constant zero value, that is, the functions have the same values, everywhere in their common region of continuity. But two functions can have the same values at a succession of isolated points, without having the same values everywhere in their common region of continuity ; in such a case the theorem does not apply, the reason being that the fundamental condition of equality over a continuous area or along a continuous line is not satisfied. COROLLARY II. A function cannot be zero over any continuous area of its region of continuity which is not infinitesimal or along any line in that region which is not infinitesimally short without being zero everywhere in its region of continuity. This corollary is deduced in the same manner as that which precedes. If, then, there be a function which is evidently not zero everywhere, we conclude that its zeros are isolated points though such points may be multiple zeros. Further, in any finite area of the region of continuity of a function that is subject to variation, there can be at most only a finite number of its zeros, when 37.] UNIFORM FUNCTION 61 no point of the boundary of the area is infinitesimally near an essential singularity. For if there were an infinite number of such points in any such region, there must be a cluster in at least one area or a succession along at least one line, infinite in number and so close as to constitute a continuous area or a continuous line where the function is everywhere zero. This would require that the function should be zero everywhere in its region of continuity, a condition excluded by the hypothesis. And it immediately follows that the points (other than those infini tesimally near an essential singularity) in a region of continuity, at which a function assumes any the same value, are isolated points ; and that only a finite number of such points occur in any finite area. 38. THEOREM II. The multiplicity m of any zero a of a function is finite provided the zero be an ordinary point of the function, which is not zero throughout its region of continuity; and the function can be expressed in the where <f> (z) is holomorphic in the vicinity of a, and a is not a zero of <£ (z). Let f(z) denote the function ; since a is a zero, we have f(a) = 0. Suppose that /'(a), f" (a), ...... vanish: in the succession of the derivatives of f, one of finite order must be reached which does not have a zero value. Otherwise, if all vanish, then the function and all its derivatives vanish at a; the expansion of f(z) in powers of z — a leads to zero as the value of f (z\ that is, the function is everywhere zero in the region of continuity, if all the derivatives vanish at a. Let, then, the wth derivative be the first in the natural succession which does not vanish at a, so that m is finite. Using Cauchy's expansion, we have (? — n\tm) ( ~ _ n\(m+\) f(z) = (Z a /« (a) + S£Za_/F* (a) + . . . J m ! J (m + 1) ! J = (z-ay*$(z\ where <£ (z) is a function that does not vanish with a and, being the quotient of a converging series by a monomial factor, is holomorphic in the immediate vicinity of a. COROLLARY I. If infinity be a zero of a function of multiplicity m and at the same time be an ordinary point of the function, then the function can be expressed in the form z~m $ f-J , where </>(-) is a function that is continuous and non-evanescent for infinitely large values of z. The result can be derived from the expansion in § 30 in the same way as the foregoing theorem from Cauchy's expansion. 62 ZEROS OF A [38. COROLLARY II. The number of zeros of a function, account being taken of their multiplicity, which occur within a finite area of the region of continuity of the function, is finite, when no point of the boundary of the area is infinitesi- mally near an essential singularity. By Corollary II. of § 37, the number of distinct zeros in the limited area is finite, and, by the foregoing theorem, the multiplicity of each is finite ; hence, when account is taken of their respective multiplicities, the total number of zeros is still finite. The result is, of course, a known result for an algebraical polynomial ; but the functions in the enunciation are not restricted to be of the type of algebraical polynomials. Note. It is important to notice, both for the Theorem and for Corollary I, that the zero is an ordinary point of the function under consideration ; the implication therefore is that the zero is a definite zero and that in the immediate vicinity of the point the function can be represented in the form P(z — a) or P [-] , the function P(a — a) or P (— ) being .always a definite \<6 / \ / zero. Instances do occur for which this condition is not satisfied. The point may not be an ordinary point, and the zero value may be an indeterminate zero ; or zero may be only one of a set of distinct values though everywhere in the vicinity the function is regular. Thus the analysis of § 13 shews that z=a is a point where the function sn - - has any number of zero values and Z CL any number of infinite values, and there is no indication that there are not also other values at the point. In such a case the preceding proposition does not apply ; there may be no limit to the order of multiplicity of the zero, and we certainly cannot infer that any finite integer m can be obtained such that (z - a)~m <j> (z) is finite at the point. Such a point is (§ 32) an essential singularity of the function. 39. THEOREM III. A multiple zero of a function is a zero of its derivative ; and the multiplicity for the derivative is less or is greater by unity according as the zero is not or is at infinity. If a be a point in the finite part of the plane which is a zero of f(z) of multiplicity n, we have /(f)-(*T.a)» + («X and therefore /' (z) = (z - a)n~l [n$ (z} + (z-a) $ (z)}. The coefficient of (z — a)n~l is holomorphic in the immediate vicinity of a and does not vanish for a ; hence a is a zero for /' (z) of decreased multiplicity 39.] UNIFORM FUNCTION If z = oo be a zero off(z) of multiplicity r, then where <£ (-) is holomorphic for very large values of z and does not vanish at \z / infinity. Therefore The coefficient of ^~r~1 is holomorphic for very large values of z, and does not vanish at infinity ; hence z=<x> is a zero off (z) of increased multiplicity r + l. Corollary I. If a function be finite at infinity, then z = oo is a zero of the first derivative of multiplicity at least two. Corollary II. If a be a finite zero off(z) of multiplicity n, we have f(z)= n #(z) f(z) ir-** fW Now a is not a zero of <J> (z) ; and therefore ^4^r is finite, continuous, uniform 9W and monogenic in the immediate vicinity of a. Hence, taking the integral of both members of the equation round a circle of centre a and of radius so small as to include no infinity and no zero, other than a, of / (z) _ and therefore no zero of $(z) — we have, by § 17 and Ex. 2, § 25, ~jT/ \ ^"^ ~ ^- /(*) 40. THEOREM IV. A function must have an infinite value for some finite or infinite value of the variable. If M be a finite maximum value of the modulus for points in the plane, then (§ 22) we have where r is the radius of an arbitrary circle of centre a, provided the whole of the circle is in the region of continuity of the function. But as the function is uniform, monogenic, finite and continuous everywhere, this radius can be increased indefinitely ; when this increase takes place, the limit of is zero and therefore /<»> (a) vanishes. This is true for all the indices 1,2,... of the derivatives. 64 INFINITIES OF A [40. Now the function can be represented at any point z in the vicinity of a by the series which degenerates, under the present hypothesis, to /(a), so that the function is everywhere constant. Hence, if a function has not an infinity somewhere in the plane, it must be a constant. The given function is not a constant; and therefore there is no finite limit to the maximum value of its modulus, that is, the function acquires an infinite value somewhere in the plane. COROLLARY I. A function must have a zero value for some finite or infinite value of the variable. For the reciprocal of a uniform monogenic analytic function is itself a uniform monogenic analytic function ; and the foregoing proposition shews that this reciprocal must have an infinite value for some value of the variable, which therefore is a zero of the function. COROLLARY II. A function must assume any assigned value at least once. COROLLARY III. Every function which is not a mere constant must have at least one singularity, either accidental or essential. For it must have an infinite value : if this be a determinate infinity, the point is an accidental singularity (§ 32) ; if it be an infinity among a set of values at the point, the point is an essential singularity (§§ 32, 33). 41. Among the infinities of a function, the simplest class is that con stituted by its accidental singularities, already defined (§ 32) by the property that, in the immediate vicinity of such a point, the reciprocal of the function is regular, the point being an ordinary (zero) point for that reciprocal. THEOREM V. A function, which has a point cfor an accidental singularity, can be expressed in the foi*m (z - c}~n (f> (z), where n is a finite positive integer and <f> (z) is a continuous function in the vicinity of c. Since c is an accidental singularity of the function f(z}, the function ^y-r / (z) is regular in the vicinity of c and is zero there (§ 32). Hence, by § 38, there is a finite limit to the multiplicity of the zero, say n (which is a positive integer), and we have where ^ (z) is uniform, monogenic and continuous in the vicinity of c and is not zero there. The reciprocal of ^ (z), say <f> (z), is also uniform, monogenic 41.] UNIFORM FUNCTION 65 and continuous in the vicinity of c, which is an ordinary point for (f> (z) ; hence we have f(z} = (Z-c)-^(z\ which proves the theorem. The finite positive integer n measures the multiplicity of the accidental singularity at c, which is sometimes said to be of multiplicity n or of order n. Another analytical expression for f(z) can be derived from that which has just been obtained. Since c is an ordinary point for <f> (z) and not a zero, this function can be expanded in a series of ascending, positive, integral powers of z — c, converging in the vicinity of c, in the form £(*) = P(*-c) = uQ + ul(z-c} + ... + un^(z-c)n-l+un(z-c)n+... = u0 + u,(z - c) + ... + un_^(z - c)71-1 + (z- c)nQ(z-c), where Q(z — c), a series of positive, integral, powers of z — c converging in the vicinity of c, is a monogenic analytic function of z. Hence we have ^ = ^» + (7^+ - +,~; + «('-')> the indicated expression for f(z), valid in the immediate vicinity of c, where Q (z — c) is uniform, finite, continuous and monogenic. COROLLARY. A function, which has z= oo for an accidental singularity of multiplicity n, can be expressed in the form _ where </>(-) is a continuous function for very large values of \z , and is not \zj zero when z = oo . It can also be expressed in the form 1 + ... + an^ z + Q (-} , \zj where Q ( - j is uniform, finite, continuous and monogenic for very large values f\»\. The derivation of the form of the function in the vicinity of an accidental singularity has been made to depend upon the form of the reciprocal of the function. Whatever be the (finite) order of that point as a zero of the reciprocal, it is assumed that other zeros of the reciprocal are not at merely infinitesimal distances from the point, that is, that other infinities of the function are not at merely infinitesimal distances from the point. Hence the accidental singularities of a function are isolated points ; and there is only a finite number of them in any limited portion of the plane. F. 5 66 INFINITIES OF A [42. 42. We can deduce a criterion which determines whether a given singu larity of a function /(f) is accidental or essential. When the point is in the finite part of the plane, say at c, and a finite positive integer n can be found such that is not infinite at c, then c is an accidental singularity. When the point is at infinity and a finite positive integer n can be found such that is not infinite when z = oc , then z = oo is an accidental singularity. If one of these conditions be not satisfied, the singularity at the point is essential. But it must not be assumed that the failure of the limitation to finiteness in the multiplicity of the accidental singularity is the only source or the complete cause of essential singularity. Since the association of a single factor with the function is effective in preventing an infinite value at the point when one of the conditions is satisfied, it is justifiable to regard the discontinuity of the function at the point as not essential and to call the singularity either non-essential or accidental (§ 82). 43. THEOREM VI. The poles of a function, that lie in the finite part of the plane, are all the poles (of increased multiplicity) of the derivatives of the function that lie in the finite part of the plane. Let c be a pole of the function f(z) of multiplicity p : then, for any point z in the vicinity of c, where </> (z) is holomorphic in the vicinity of c, and does not vanish for z = c. Then we have f'(2) = (z~ c)~p $' (z) ~ P (2 ~ c) p 1 $ W = (z-c)-P-*{(z-c)<j>'(z)-p<}>(z)} where % (z) is holomorphic in the vicinity of c, and does not vanish for z = c. Hence c is a pole of/' (z) of multiplicity ^9 + 1. Similarly it can be shewn to be a pole of /(r) (z) of multiplicity p + r. This proves that all the poles of f(z) in the finite part of the plane are poles of its derivatives. It remains to prove that a derivative cannot have a pole which the original function does not also possess. Let a be a pole off'(z) of multiplicity m : then, in the vicinity of a,f'(z) can be expressed in the form 43.] UNIFORM FUNCTION £7 where ^ (z) is holomorphic in the vicinity of a and does not vanish for z = a Thus and therefore f (*) = - . + j_ , y v ' JlV^ •<*-«)*"* so that, integrating, we have f(z}= *(«) _*>) m 0 - a)™-1 (m - 1) 0 - a)™-2 that is, a is a pole of/0). An apparent exception occurs in the case when m is unity: for then we have the integral of which leads to f(z} = ^ (a) log (z - a) + . . . , so that/0) is no longer uniform, contrary to hypothesis. Hence a derivative cannot have a simple pole in the finite part of the plane ; and so the exception is excluded. The theorem is thus proved. COROLLARY I. The rth derivative of a function cannot have a pole in the finite part of the plane of multiplicity less than r + 1. COROLLARY II. If c be a pole of f (z) of any order of multiplicity ^ and if f(r] (z) be expressed in the form , _ Oi__ » / _. _\., _!_*• _ 1 I ••••••* (Z - CY+T (Z- there are no terms in this expression with the indices - 1, - 2, ...... , - r. COROLLARY III. If c be a pole of/ (z) of multiplicity p, we have = f(z) z-c~* 4>(z)' where $ (z) is a holomorphic function that does not vanish for z = c, so that <£' 0) • -T-/JN is a holomorphic function in the vicinity of c. Taking the integral of f'(z) -j-j~\ round a circle, with c for centre, with radius so small as to exclude all other poles or zeros of the function f (z), we have 5—2 (}8 INFINITIES OF A [43. COROLLARY IV. If a simple closed curve include a number N of zeros of a uniform function f (z) and a number P of its poles, in both of which numbers account is taken of possible multiplicity, and if the curve contain no essential singularity of the function, then the integral being taken round the curve. f (z) The only infinities of the function ' ^i within the curve are the zeros j(z) and the poles of / (z). Round each of these draw a circle of radius so small as to include it but no other infinity ; then, by Cor. II. § 18, the integral round the closed curve is the sum of the values when taken round these circles. By the Corollary II. § 39 and by the preceding Corollary III., the sum of these values is = 2w — %> = N-P. It is easy to infer the known theorem that the number of roots of an algebraical polynomial of order n is n, as well as the further result that 2^ (N - P) is the variation of the argument of / (z) as z describes the closed curve in a positive sense. Ex. Prove that, if F(z) be holomorphic over an area, of simple contour, which con tains roots «!, «2,... of multiplicity m» m2,... and poles cx, c2)... of multiplicity p^ p2J... respectively of a function f(z) which has no other singularities within the contour, then the integral being taken round the contour. In particular, if the contour contains a single simple root a and no singularity, then that root is given by the integral being taken as before. (Laurent.) 44. THEOREM VII. If infinity be a pole of f (z), it is also a pole of f (z) only when it is a multiple pole of f (z). Let the multiplicity of the pole for f (z) be ?i; then for very large values of z we have /(*)-*•*£), where <j> is holomorphic for very large values of z and does not vanish at infinity ; hence A«)**" •*-*'• 44.] UNIFORM FUNCTION 69 The coefficient of zn~* is holomorphic for very large values of z and does not vanish at infinity ; hence infinity is a pole of/' (z} of multiplicity n — 1. If n be unity, so that infinity is a simple pole of / (z), then it is not a pole of/' (2); the derivative is then finite at infinity. 45. THEOREM VIII. A function, which has no singularity in a finite part of the plane, and has z = oo for a pole, is an algebraical polynomial. Let n, necessarily a finite integer, be the order of multiplicity of the pole at infinity : then the function / (z) can be expressed in the form 1 + ...... +an^z + Q - , \zJ where Q (- J is a holomorphic function for very large values of z, and is finite (or zero) when z is infinite. Now the first n terms of the series constitute a function which has no singularities in the finite part of the plane : and / (z) has no singularities in that part of the plane. Hence Q ( - J has no singularities in the finite part of the plane : it is finite for infinite values of z. It thus can never have an infinite value: and it is therefore merely a constant, say an. Then / (z) = a,zn + a^-1 + ...... + an^z + an, a polynomial of degree equal to the multiplicity of the pole at infinity, supposed to be the only pole of the function. 46. The above result may be obtained in the following manner. Since z = GO is a pole of multiplicity n, the limit of z~nf (z} is not infinite when z = oo . Now in any finite part of the plane the function is everywhere finite, so that we can use the expansion where £ = *'""> dt ''+l t-z' the integral being taken round a circle of any radius r enclosing the point z and having its centre at the origin. As the subject of integration is finite everywhere along the circumference, we have, by Darboux's expression in (IV.) S 14, T»i T _ z where r is some point on the circumference and X is a quantity of modulus not greater than unity. 70 TRANSCENDENTAL AND [46. Let T = reia- ; then X . fM "• 71-4-1 °flii »/ \ / '?* rn r f(T\ By definition, the limit of n as T (and therefore r) becomes infinitely (£ -\—1 1 -- e~ai } is unity. r J Since \ is not greater than unity, the limit of \jr in the same case is zero ; hence with indefinite increase of r, the limit of R is zero and so shewing as before that/(^) is an algebraical polynomial. 47. As the quantity n is necessarily a positive integer*, there are two distinct classes of functions discriminated by the magnitude of n. The first (and the simpler) is that for which n has a finite value. The polynomial then contains only a finite number of terms, each with a positive integral index ; and the function is then a rational, integral, algebraical polynomial of degree n. The second (and the more extensive, as significant functions) is that for which n has an infinite value. The point z = oo is not a pole, for then the function does not satisfy the test of § 42 : it is an essential singularity of the function, which is expansible in an infinite converging series of positive integral powers. To functions of this class the general term transcendental is applied. The number of zeros of a function of the former class is known : it is equal to the degree of the function. It has been proved that the zeros of a transcendental function are isolated points, occurring necessarily in finite number in any finite part of the region of continuity of the function, no point on the boundary of the part being infinitesimally near an essential singularity ; but no test has been assigned for the determination of the total number of zeros of a function in an infinite part of the region of con tinuity. Again, when the zeros of a polynomial are given, a product-expression can at once be obtained that will represent its analytical value. Also we know that, if a be a zero of any uniform analytic function of multiplicity n, the function can be represented in the vicinity of a by the expression (x-a}n<t>(z\ where <£ (z) is holomorphic in the vicinity of a. The other zeros of the function are zeros of <f> (z) ; this process of modification in the expression * It is unnecessary to consider the zero value of n, for the function is then a polynomial of order zero, that is, it is a constant. 47.] ALGEBRAICAL UNIFORM FUNCTIONS 71 can be continued for successive zeros so long as the number of zeros taken account of is limited. But when the number of zeros is unlimited, then the inferred product-expression for the original function is not necessarily a converging product; and thus the question of the formal factorisation of a transcendental function arises. 48. THEOREM IX. A function, all the singularities of which are accid ental, is a rational, algebraical, meromorphic function. Since all the singularities are accidental, each must be of finite multiplicity ; and therefore infinity, if an accidental singularity, is of finite multiplicity. All the other poles are in the finite part of the plane ; they are isolated points and therefore only finite in number, so that the total number of distinct poles is finite and each is of finite order. Let them be «!, a2, ...... , a^ of orders m1} m2, ...... , m^ respectively : let m be the order of the pole at infinity: and let the poles be arranged in the sequence of decreasing moduli such that [aj > aF_! > ...... >|&i|- Then, since infinity is a pole of order m, we have / 0) = amzm + a^z™-1 + ...... + a^z + /„ <», where /„ (z) is not infinite for infinite values of z. Now the polynomial m Sttj^ is not infinite for any finite value of z ; hence f0 (z) is infinite for all i = l the finite infinities of f (z) and in the same way, that is, the function f0(z) has «!, ...... , a^ for its poles and it has no other singularities. Again, since «M is a finite pole of multiplicity WM, we have where fi(z) is not infinite for z = all and, as f0(z) is not infinite for z=<x> , evidently f^ (z) is not infinite for z = oo . Hence the singularities of f^ (z) are merely the poles a1} ...... , aF_i ; and these are all its singularities. Proceeding in this manner for the singularities in succession, we ultimately reach a function f^ (z) which has only one pole a^ and no other singularity, so that k k where g (z) is not infinite for z = a^ But the function f^(z) is infinite only for 2 = 0,!, and therefore g (2) has no infinity. Hence g (z} is only a constant, say k0 : thus 9 (*} = ^o- Combining all these results we have a, finite number of series to add together: and the result is that 72 UNIFORM [48. where g1 (z) is the series k0 + a-^z + + amzm, and \ I is the sum of the finite number of fractions. Evidently gs (z) is the product {z — Oi)m> (z — a2)ma (z — aM)mfx ; and g» (z) is at most of degree If F (z} denote g1 (z} g3 (z) + g^ (z), the form of / (z) is </.(*)' that is, f (z) is a rational, algebraical, meromorphic function. It is evident that, when the function is thus expressed as an algebraical fraction, the degree of F (z) is the sum of the multiplicities of all the poles when infinity is a pole. COROLLARY I. A function, all the singularities of which are accidental, has as many zeros as it has accidental singularities in the plane. If z = oo be a pole, then it follows that, because f(z) can be expressed in the form it has as many zeros as F(z), unless one such should be also a zero of g^(z). But the zeros of g3(z) are known, and no one of them is a zero of F(z), on account of the form of f(z} when it is expressed in partial fractions. Hence the number of zeros off(z) is equal to the degree of F(z}, that is, it is equal to the number of poles off(z}. If 2=00 be not a pole, two cases are possible; (i) the function f (z) may be finite for z = oo , or (ii) it may be zero for z = oo . In the former case, the number of zeros is, as before, equal to the degree of F (z), that is, it is equal to the number of infinities. In the latter case, if the degree of the numerator F (z) be K less than that of the denominator gs (z), then z = oo is a zero of multiplicity K ; and it follows that the number of zeros is equal to the degree of the numerator together with K, so that their number is the same as the number of accidental singularities. COROLLARY II. At the beginning of the proof of the theorem of the present section, it is proved that a function, all the singularities of which are accidental, has only a finite number of such singularities. Hence, by the preceding Corollary, such a function can have only a finite number of zeros. If, therefore, the number of zeros of a function be infinite, the function must have at least one essential singularity. 48.] ALGEBRAICAL FUNCTIONS 73 COROLLARY III. When a uniform analytic function has no essential singularity, if the (finite) number of its poles, say clv.., cm, be m, no one of them being at z = oo , and if the number of its zeros, say aly..., am, be also m, no one of them being at z = oo , then the function is „ n * a r=l \Z - CT except possibly as to a constant factor. When z = oo is a zero of order n, so that the function has m — n zeros, say «i, a2,..., in the finite part of the plane, the form of the function is m-n II (z — ar) r=l r=l and, when z = <x> is a pole of order p, so that the function has m - p poles, say cl} c.2>..., in the finite part of the plane, the form of the function is II (Z - Or) r=l _ m-p ~ COROLLARY IV. All the singularities of rational algebraical meromorphic functions are accidental. CHAPTER V. TRANSCENDENTAL INTEGRAL FUNCTIONS. 49. WE now proceed to consider the properties of uniform functions which have essential singularities. The simplest instance of the occurrence of such a function has already been referred to in § 42 ; the function has no singularity except at z = oo , and that value is an essential singularity solely through the failure of the limitation to finiteness that would render the singularity accidental. The function is then an integral function of transcendental character ; and it is analytically represented (§ 26) by G (z) an infinite series in positive powers of z, which converges everywhere in the finite part of the plane and acquires an infinite value at infinity alone. The preceding investigations shew that uniform functions, all the singu larities of which are accidental, are rational algebraical functions — their character being completely determined by their uniformity and the accidental nature of their singularities, and that among such functions having the same accidental singularities the discrimination is made, save as to a constant factor, by means of their zeros. Hence the zeros and the accidental singularities of a rational algebraical function determine, save as to a constant factor, an expression of the function which is valid for the whole plane. A question therefore arises how far the zeros and the singularities of a transcendental function determine the analytical expression of the function for the whole plane. 50. We shall consider first how far the discrimination of transcendental integral functions, which have no infinite value except for z = oc , is effected by means of their zeros*. * The following investigations are based upon the famous memoir by Weierstrass, " Zur Theorie der eindeutigen analytischen Functionen," published in 187G : it is included, pp. 1 — 52, in the Abhandlungen aus der Functioiienlehre (Berlin, 1886). In connection with the product-expression of a transcendental function, Cayley, " Memoire sur les fonctions doublement periodiques," Liouville, t. x, (1845), pp. 385 — 420, or Collected Works, vol. i, pp. 156 — 182, should be consulted. 50.] CONVERGING INFINITE PRODUCTS 75 Let the zeros aly a2, a3,... be arranged in order of increasing moduli; a finite number of terms in the series may have the same value so as to allow for the existence of a multiple zero at any point. After the results stated 47, it will be assumed that the number of zeros is infinite ; that, n subject to limited repetition, they are isolated points ; and, in the present chapter, that, as n increases indefinitely, the limit of \an\ is infinity. And it will be assumed that at\ > 0, so that the origin is temporarily excluded from the series of zeros. Let z be any point in the finite part of the plane. Then only a limited number of the zeros can lie within and on a circle centre the origin and radius equal to \z\ ; let these be a]5 a2,..., afc_1} and let ar denote any one of the other zeros. We proceed to form the infinite product of quantities ur, where ur denotes and gr is a rational integral function of z which, being subject to choice, will be chosen so as to make the infinite product converge everywhere in the plane. We have 00 \ w=l a series which converges because \z < \ar\. Now let ffr = then «> 1 / ^ \n i v -1- / •* \ logi<r = - 2 -f£J , »j = S »4 \**rr and therefore Hence •-— " if the expression on the right-hand side be finite, that is, if the series oo ce I / _ \ n 2 S -(-) r=ftw=«^ \flrf converge unconditionally. Denoting the modulus of this series by M, we have z a,. 00 00 1 M < 2 2 - r-k n=s M SO that sM< S 2 r=k n=s 7G WEIERSTRASS'S CONVERGING [50. whence since 1 - — is the smallest of the denominators in terms of the last «* sum, we have sM\l- z [ < 00 Z 8 1 «& j r=k ar • I l • *-l If, as is not infrequently the case, there be any finite integer s for which (and therefore for all greater indices) the series 2 1 Is ' 00 and therefore the series 2 \ar\-s, converges, we choose s to be that least r=k integer. The value of M then is finite for all finite values of z ; the series oo co T / ~\n 2 2 - - n r=k converges unconditionally and therefore is a converging product when Let the finite product A-l (/ f n |(i-- m=l l\ am be associated as a factor with the foregoing infinite converging product. Then the expression oo ( f 2 \ 2 T-=I (\ ar/ is an infinite product, converging uniformly and unconditionally for all finite 00 values of z, provided the finite integer s be such as to make the series 2 converge uniformly and unconditionally. Since the product converges uniformly and unconditionally, no product constructed from its factors ur, say from all but one of them, can be infinite. Now the factor "5?i/£-Y \ ?L\en=\n\am) vanishes for z = am; hence f(z) vanishes for z = am. Thus the function, evidently uniform after what has been proved, has the assigned points Oj, a2)... and no others for its zeros. 50.] INFINITE PRODUCT 77 Further, z = oo is an essential singularity of the function ; for it is an essential singularity of each of the factors on account of the exponential element in the factor. 51. But it may happen that no finite integer s can be found which will make the series 00 r=l converge*. We then proceed as follows. Instead of having the same index s throughout the series, we associate with every zero ar an integer mr chosen so as to make the series n=l @"n \Q"n a converging series. To obtain these integers, we take any series of decreasing real positive quantities e, e1} e2,..., such that (i) e is less than unity and (ii) they form an unconditionally converging series ; and we choose integers ftir such that These integers make the foregoing series of moduli converge. For, neglecting the limited number of terms for which \z\^ a\, and taking e such that z we have for all succeeding terms and therefore ar Hence, except for the first k — 1 terms, the sum of which is finite, we have n=k which is finite because the series ... converges. Hence the series n=l s a converging series. * For instance, there is no finite integer s that can make the infinite series (log 2)-' + (log 3)- + (log 4)- + . . . converge. This series is given in illustration by Hermite, Cours a la faculte des Sciences (4mc ed. 1891), p. 86. 78 WEIERSTRASS'S CONVERGING [51. Just as in the preceding case a special expression was formed to serve as a typical factor in the infinite product, we now form a similar expression for the same purpose. Evidently 1 - a; = ei<* a-*) = e if \x\ < 1. Forming a function E (x, m) denned by the equation m xr S - E (x, m)=(l-x)e r=1 r , we have E (x, m) = In the preceding case it was possible to choose the integer m so that it should be the same for all the factors of the infinite product, which was 0 ultimately proved to converge. Now, we take x = — and associate mn as the corresponding value of m. Hence, if /(*) = where < \z < |ttjfc|, we have n=k - s s The infinite product represented by f(z) will converge if the double series in the exponential be a converging series. Denoting the double series by S, we have \S\<* 2 2^* 2 n=kr=l r+mn < 2 n—k 1+TOM 1 4£ \an on effecting the summation for r. Let A be the value of 1 — all the remaining values of n we have 1 z ! - ; then for — >>A, and so n=/fc This series converges; hence for finite values of z\ the value of \S\ is finite, so that S is a converging series. Hence it follows that f(z) is an 51.] INFINITE PRODUCT 79 unconditionally converging product. We now associate with f(z) as factors the k — I functions for i= 1, 2,..., k—1; their number being finite, their product is finite and therefore the modified infinite product still converges. We thus have an unconditionally converging product. Since the product G (z) converges unconditionally, no product constructed from its factors E, say from all but one of them, can be infinite. The factor vanishes for the value z = an and only for this value ; hence G (z) vanishes for z = an. It therefore appears that G(z) has the assigned points a1} a.,, a3, ... and no others for its zeros ; and from the existence of the exponential in each of the factors it follows that z = oo is an essential singularity of the factor and therefore it is an essential singularity of the function. Denoting the series in the exponential by gn (z\ so that mn 1 / ~ \ r *<*>-£?(£)• 71 / z \ i-. Z\ we have A — , mn = 1 e^' ; \an / V aJ and therefore the function obtained is ; G (z)= H \(l — — ] eg«(zl n = l (\ Q"n,l The series gn usually contains only a limited number of terms ; when the number of terms increases without limit, it is only with indefinite increase of | an | and the series is then a converging series. It should be noted that the factors of the infinite product G (z) are the expressions E no one of which, for the purposes of the product, is resoluble into factors that can be distributed and recombined with similarly obtained factors from other expressions E; there is no guarantee that the product of the factors, if so resolved, would converge uniformly and unconditionally, and it is to secure such convergence that the expressions E have been constructed. It was assumed, merely for temporary convenience, that the origin was not a zero of the required function ; there obviously could not be a factor of exactly the same form as the factors E if a were the origin. 80 TRANSCENDENTAL INTEGRAL FUNCTION [51. If, however, the origin were a zero of order X, we should have merely to associate a factor ZK with the function already constructed. We thus obtain Weierstrass's theorem : — It is possible to construct a transcendental integral function such that it shall have infinity as its only essential singularity and have the origin (of multiplicity X), a^, az, a3, ... as zeros ; and such a function is 00 ( / z\ ZK n ui — U^ n=i where gn(z) is a rational, integral, algebraical function of z, the form of which is dependent upon the law of succession of the zeros. 52. But, unlike uniform functions with only accidental singularities, the function is not unique : there are an unlimited number of transcendental integral functions with the same series of zeros and infinity as the sole essential singularity, a theorem also due to Weierstrass. For, if G! (z) and G (z) be two transcendental, integral functions with the same series of zeros in the same multiplicity, and z = oo as their only essential singularity, then G(z} is a function with no zeros and no infinities in the finite part of the plane. Denoting it by £r2, then 1 ^ <72 dz is a function which, in the finite part of the plane, has no infinities; and therefore it can be expanded in the form a series converging everywhere in the finite part of the plane. Choosing a constant C0 so that 6r2 (0) = e*7", we have on integration where g(z) = C0 and g (z) is finite everywhere in the finite part of the plane. Hence it follows that, ifg(z) denote any integral function of z which is finite everywhere in the finite part of the plane, and if G (z) be some transcendental integral function with a given series of zeros and z= oo as its sole essential singularity, all transcendental integral functions with that series of zeros and z= <x> as the sole essential singularity are included in the form £(*)«*». COROLLARY I. A function which has no zeros in the finite part of the plane, no accidental singularities and z=<x> for its sole essential singularity is necessarily of the form 52.] AS AN INFINITE PRODUCT 81 where g (z) is an integral function of z finite everywhere in the finite part of the plane. COROLLARY II. Every transcendental function, which has the same zeros in the same multiplicity as an algebraical polynomial A (z) — the number, therefore, being necessarily finite — , ivhich has no accidental singularities and has z = oo for its sole essential singularity, can be expressed in the form A (z) COROLLARY III. Every function, which has an assigned series of zeros and an assigned series of poles and has z = oo for its sole essential singu larity, is of the form where the zeros of G0(z) are the assigned zeros and the zeros of Gp(z) are the assigned poles. For if Op (z) be any transcendental integral function, constructed as in the proposition, which has as its zeros the poles of the required function in the assigned multiplicity, the most general form of that function is 0p(*)e*», where h (z) is integral. Hence, if the most general form of function which has those zeros for its poles be denoted by f(z), we have f(z)Gp(z)e^ as a function with no poles, with infinity as its sole essential singularity, and with the assigned series of zeros. But if G0 (z) be any transcendental integral function with the assigned zeros as its zeros, the most general form of function with those zeros is and so f(z) Gp (z) eh ® = G0 (z) e° & , whence / (z) = ?$1 effW, Lrp (z) in which g (z) denotes g (z) — h (z). If the number of zeros be finite, we evidently may take G0(z) as the algebraical polynomial with those zeros as its only zeros. If the number of poles be finite, we evidently may take Gp(z) as the algebraical polynomial with those poles as its only zeros. And, lastly, if a function have a finite number of zeros, a finite number of accidental singularities and 2=00 as its sole essential singularity, it can be expressed in the form F. 82 PRIMARY [52. where P and Q are rational integral polynomials. This is valid even though the number of assigned zeros be not the same as the number of assigned poles ; the sole effect of the inequality of these numbers is to complicate the character of the essential singularity at infinity. 53. It follows from what has been proved that any uniform function, having z = <x> for its sole essential singularity and any number of assigned zeros, can be expressed as a product of expressions of the form a Such a quantity is called* a primary factor of the function. It has also been proved that : — (i) If there be no zero an, the primary factor has the form (ii) The exponential index gn (z) may be zero for individual primary factors, though the number of such factors must, at the utmost, be finite f. (iii) The factor takes the form z when the origin is a zero. Hence we have the theorem, due to Weierstrass : — Every uniform integral function of z can be expressed as a product of primary factors, each of the form (kz + I) e3W, where g(z) is an appropriate integral function of z vanishing with z and where k, I are constants. In particular factors, g (z) may vanish ; and either k or I, but not both k and I, may vanish with or without a non-vanishing exponential index g(z). 54. It thus appears that an essential distinction between transcendental integral functions is constituted by the aggregate of their zeros : and we may conveniently consider that all such functions are substantially the same when they have the same zeros. There are a few very simple sets of functions, thus discriminated by their zeros: of each set only one member will be given, and the factor e^(z}, which makes the variation among the members of the same set, will be neglected for the present. Moreover, it will be assumed that the zeros are isolated points. I. There may be a finite number of zeros ; the simplest function is then an algebraical polynomial. * Weierstrass's term is Prim/unction, I.e., p. 15. t Unless the class (§ 59) be zero, when the index is zero for all the factors. 54.] FACTORS 83 II. There may be a singly-infinite system of zeros. Various functions will be obtained, according to the law of distribution of the zeros. Thus let them be distributed according to a law of simple arithmetic progression along a given line. If a be a zero, co a quantity such that co \ is the distance between two zeros and arg. co is the inclination of the line, we have a + mco, for integer values of m from - oo to + oo , as the expression of the series of the zeros. Without loss of generality we may take a at the origin — this is merely a change of origin of coordinates — and the origin is then a simple zero : the zeros are given by mco, for integer values of m from — oo to + oo . Now 2 — - = - 2 — is a diverging series ; but an integer s — the lowest value is s = 2 — can be found for which the series S I - ] converges uni- \mcoj formly and unconditionally. Taking s = 2, we have , . '-1 1 / z \n z ffm (z) = 2 - — = — , »=i n vW m™ so that the primary factor of the present function is Z \ --- ) mco/ m<a e and therefore, by § 52, the product /«-,SJ(i- *-) -oo (\ mcoj converges uniformly and unconditionally for all finite values of z. The term corresponding to m = 0 is to be omitted from the product ; and it is unnecessary to assume that the numerical value of the positive infinity for m is the same as that of the negative infinity for m. If, however, the latter assumption be adopted, the expression can be changed into the ordinary product-expression for a sine, by combining the primary factors due to values of m that arc equal and opposite : in fact, then co . TTZ = - - sin — . 7T CO This example is sufficient to shew the importance of the exponential term in the primary factor. If the product be formed exactly as for an algebraical polynomial, then the function is z n in the limit when both p and q are infinite. But this is known* to be - ) - sin — . 77 0) * Hobson's Trigonometry, § 287. 6—2 84 PRIMARY [54. Another illustration is afforded by Gauss's II-function, which is the limit when k is infinite of 1.2.3 ...... k («+!) (0+2) ...... (z+k) This is transformed by Gauss* into the reciprocal of the expression that is, of (1 +*) jj {(l +^) e "2l°g the primary factors of which have the same characteristic form as in the preceding investigation, though not the same literal form. It is chiefly for convenience that the index of the exponential part of the primary t-l 1/2 \n factor is taken, in § 50, in the form 2 - ( — ) . With equal effectiveness it may be n=l % \~^T / »-l 1 taken in the form 2 - br nzn. provided the series ' r=k «=i n converge uniformly and unconditionally. Ex. 1. Prove that each of the products form=+l, ±3, +5, ...... to infinity, and the term for n = Q being excluded from the latter product, converges uniformly and uncon ditionally and that each of them is equal to cos z. (Hermite and Weyr.) Ex. 2. Prove that, if the zeros of a transcendental integral function be given by the series 0) +&>, ±4w, +9cB, ...... to infinity, the simplest of the set of functions thereby determined can be expressed in the form ( fz\*\ , (. fz\*\ sm X?r I - }- sin -UTT - ) }- . I W ) ( W J Ex. 3. Construct the set of transcendental integral functions which have in common the scries of zeros determined by the law m2a>l + 2m<a2 + a>3 for all integral values of m between - oo and + oo ; and express the simplest of the set in terms of circular functions, j 55. The law of distribution of the zeros, next in importance and sub stantially next in point of simplicity, is that in which the zeros form a doubly- infinite double arithmetic progression, the points being the oo 2 intersections of one infinite system of equidistant parallel straight lines with another infinite system of equidistant parallel straight lines. The origin may, without loss of generality, be taken as one of the zeros. If a) be the coordinate of the nearest zero along the line of one system passing through the origin, and &>' be the coordinate of the nearest zero along * Ges. Wcrke, t. Hi, p. 145; the example is quoted in this connection by Weierstrass, I.e., ! p. 15. 55.] FACTIOUS 85 the line of the other system passing through the origin, then the complete series of zeros is given by fl = mw + mm, for all integral values of m and all integral values of ni between — <x> and + oo . The system of points may be regarded as doubly -periodic, having &> arid &>' for periods. It must be assumed that the two systems of lines intersect. Other wise, w and to' would have the same argument and their ratio would be a real quantity, say a ; and then ft — = m + m a. CO Whether a be commensurable or incommensurable, the number of pairs of integers, for which m + in' a. is zero or may be made less than any small quantity 8, is infinite ; and in either case we should have the origin a zero for each such pair, that is, altogether the origin would be a zero of infinite multiplicity. This property of a function is to be considered as excluded, for it would make the origin an essential singularity instead of, as required, an ordinary point of the transcendental integral function. Hence the ratio of the quantities w and w' is not real. 56. For the construction of the primary factor, it is necessary to render the series converging, by appropriate choice of integers sm>m. It is found to be possible to choose an integer s to be the same for every term of the series, corresponding to the simpler case of the general investigation, given in § 50. As a matter of fact, the series diverges for s = I (we have not made any assumption that the positive and the negative infinities for m are numerically equal, nor similarly as to m') ; the series converges for s = 2, but its value depends upon the relative values of the infinities for m and m'; and s = 3 is the lowest integral value for which, as for all greater values, the series converges uniformly and unconditionally. There are various ways of proving the uniform and unconditional conver gence of the series 2ft~M when /* > 2 : the following proof is based upon a general method due to Eisenstein*. »I=QO n=oo First, the series S 2 (m2 + n*)~* converges uniformly and uricondi- m= — «> n= -oo tionally, if /j,> 1. Let the series be arranged in partial series : for this purpose, Crelle, t. xxxv, (1847), p. 161 ; a geometrical exposition is given by Halphen, Traite des fonctions elliptiques, t. i, pp. 358 — 362. 86 WEIERSTRASS'S FUNCTION AS [56. we choose integers k and I, and include in each such partial series all the terms which satisfy the inequalities m ^ 2*+1, so that the number of values of m is 2* and the number of values of n is 2*. Then, if k + I = %K, we have so that each term in the partial series ^ ^- . The number of terms in the ^" J* partial series is 2fc . 2*, that is, 22K : so that the sum of the terms in the partial series is Take the upper limit of k and I to be p, ultimately to be made infinite. Then the sum of all the partial series is which, when p = oo , is a finite quantity if p > 1. Next, let (a = a. + /3i, «' = 7 + Si, so that ft = mw + nay' = ma + ny + i (m{3 + n8) ; hence, if 6 =• ma. + nj, (j> = m(3 + n$, we have | ft 2 = fr + </>2. Now take integers r and s such that r<0<r + \, s<(jxs + ~L. The number of terms ft satisfying these conditions is definitely finite and is independent of m and n. For since m(«S — n a - and a8 — (3y does not vanish because o>'/a> is not purely real, the number of values of in is the integral part of (r + 1)8 — sy a.8 — fiy less the integral part of r8 — (s + 1 ) 7 a.8 — fly that is, it is the integral part of (7 + 8)/(«8 — #7). Similarly, the number of values of n is the integral part of (a + /3)/(aS - j3j). Let the product of the 56.] A DOUBLY-INFINITE PRODUCT 87 last two integers be q ; then the number of terms fl satisfying the in equalities is q. Then 22 1 ft \~* = 22 (&> + p)~* < q 22 (r2 + s2)-'*, which, by the preceding result, is finite when yu,> 1. Hence 22 (mco + m'(»)'}~-»- converges uniformly and unconditionally when //, > 1 ; and therefore the least value of s, an integer for which 22 (mco + m'co')~s converges uniformly and unconditionally, is 3. The series 22(?tto) + m'<»')~2 has a finite sum, the value of which depends* upon the infinite limits for the summation with regard to m and m'. This dependence is inconvenient and it is therefore excluded in view of our present purpose. Ex. Prove in the same manner that the series the multiple summation extending over all integers mlt m2, ...... , mn between — oo and + oo , converges uniformly and unconditionally if 2/j.>n. (Eiseustein.) 57. Returning now to the construction of the transcendental integral function the zeros of which are the various points H, we use the preceding result in connection with § 50 to form the general primary factor. Since s = 3, we have s-l and therefore the primary factor is Moreover, the origin is a simple zero. Hence, denoting the required function by a (z), we have 00 °° <r(z) = zU H — 00 -00 as a transcendental integral function which, since the product converges uni formly and unconditionally for all finite values of z, exists and has a finite value everywhere in the finite part of the plane; the quantity O denotes mco + mV, and the double product is taken for all values of m and of m between — oo and + oo , simultaneous zero values alone being excluded. This function will be called Weierstrass's o-function ; it is of importance in the theory of doubly-periodic functions which will be discussed in Chapter XL * See a paper by the author, Quart. Journ. of Math., vol. xxi, (1886), pp. 261—280. 88 PRIMARY FACTORS [57. Ex. If the doubly-infinite series of zeros be the points given by Q = m2^ + 2wm&>2 + «2o>3, wi> W2) W3 being such complex constants that i2 does not vanish for real values of m and n, then the series 2 2 Q-* converges for s = 2. The primary factor is thus and the simplest transcendental integral function having the assigned zeros is The actual points that are the zeros are the intersections of two infinite systems of parabolas. 58. One more result — of a negative character — will be adduced in this connection. We have dealt with the case in which the system of zeros is a singly-infinite arithmetical progression of points along one straight line and with the case in which the system of zeros is a doubly-infinite arithmetical progression of points along two different straight lines : it is easy to see that a uniform transcendental integral function cannot exist with a triply -infinite arithmetical progression of points for zeros. A triply-infinite arithmetical progression of points would be represented by all the possible values of for all possible integer values for p1} p.,, p3 between — oo and + oc , where no two of the arguments of the complex constants flj, H2, O3 are equal. Let tlr = o)r + i(or', (r = 1, 2, 3) ; then, as will be proved (§ 107) in connection with a later proposition, it is possible* — and possible in an unlimited number of ways — to determine integers plt p-2,ps so that, save as to infinitesimal quantities, Pi _ _ £2 ___ PS all the denominators in which equations differ from zero on account of the fact that no two arguments of the three quantities fl1} H2, Ha are equal. For each such set of determined integers we have &.Qi+p&+p»to» zero or infinitesimal, so that the origin is a zero of unlimited multiplicity or, in other words, there is a space at the origin containing an unlimited number of zeros. In either case the origin is an essential singularity, contrary to * Jacobi, Oes. Werke, t. ii, p. 27. 58.] CLASS OF A FUNCTION 89 the hypothesis that the only essential singularity is for z — oo ; and hence a uniform transcendental function cannot exist having a triply-infinite arith metical succession of zeros. 59. In effecting the formation of a transcendental integral function by means of its primary factors, it was seen that the expression of the primary factor depends upon the values of the integers which make a converging series. Moreover, the primary factors are not unique in form, because any finite number of terms of the proper form can be added to the exponential index in and such terms will only the more effectively secure the convergence of the infinite product. But there is a lower limit to the removal of terms with the highest exponents from the index of the exponential ; for there are, in general, minimum values for the integers m1} m»,..., below which these integers can not be reduced, if the convergence of the product is to be secured. The simplest case, in which the exponential must be retained in the primary factor in order to secure the convergence of the infinite product, is that discussed in § 50, viz., when the integers ml, w2)... are equal to one another. Let m denote this common value for a given function, and let m be the least integer effective for the purpose : the function is then said* to be of class m, and the condition that it should be of class m is that the integer m be the least integer to make the series converge uniformly and unconditionally, the constants a being the zeros of the function. Thus algebraical polynomials are of class 0 ; the circular functions sin z and cos z are of class 1 ; Wcierstrass's o--function, and the Jacobian elliptic function sn z are of class 2, and so on : but in .no one of these classes do the functions mentioned constitute the whole of the functions of that class. 60. One or two of the simpler properties of an aggregate of transcen dental integral functions of the same class can easily be obtained. Let a function f(z), of class n, have a zero of order r at the origin and * The French word is genre ; the Italian is genere. Laguerre (see references on p. 92) appears to have been the first to discuss the class of transcendental integral functions. 90 CLASS-PROPERTIES OF [60. have «!, a2)... for its other zeros, arranged in order of increasing moduli. Then, by § 50, the function /O) can be expressed in the form (*)=' M 1 / £\8 where </; (V) denotes the series 2 -f— 1 and G(z) must be properly deter mined to secure the equality. Now the series is one which converges uniformly for all values of z that do not coincide with one of the points a, that is, with one of the zeros of the original function. For the sum of the series of the moduli of its terms is 1 Let d be the least of the quantities 1 , necessarily non-evanescent be cause z does not coincide with any of the points a ; then the sum of the series IS 1 which is a converging series since the function is of class n. Hence the series of moduli converges and therefore the original series converges ; let it be denoted by S (z), so that 1 =2 We have Each step of this process is reversible in all cases in which the original pro- f (z\ duct converges; if, therefore, it can be shewn of a function f(z) that -rr4 takes this form, the function is thereby proved to be of class n. If there be no zero at the origin, the term - is absent. CO.] TRANSCENDENTAL INTEGRAL JUNCTIONS 91 If the exponential factor G(z) be a constant so that G' (z) is zero, the function /(.z) is said to be a simple function of class n. 61. There are one or two criteria to determine the class of a function : the simplest of them is contained in the following proposition, due to Laguerre*. If, as z tends to the value <x> , a very great value of z can be found for f'(z\ which the limit of z~n --jr\ , where f (z) is a transcendental, integral function, J\z) tends uniformly to the value zero, then f (z} is of class n. Take a circle centre the origin and radius R, equal to this value of \z\\ then, by § 24, II., the integral f'(t) dt JL/lo! SvtJ *»/(*) taken round the circle, is zero when R becomes indefinitely great. But the value of the integral is, by the Corollary in § 20, ' (t) 6A + !_ f<*> J./'_(0 Jfc_ _L y ( 27ri J V- f(t) t-z 2-n-i <=1 J tn f(fi t-Z 2-7TI J tn f(t) t-Z 2lri i=i J tn f(t} t-z' taken round small circles enclosing the origin, the point z, and the points a,i, which are the infinities of the subject of integration; the origin being supposed a zero of /(t) of multiplicity r. 1 f» !/'(*) dt ._!/'(*) JMOW tnf(t}t-Z Znf(2}' dt I I »/ \^ /• Shr», 1 fWlf(t iriJ «"/(0 L f<0> 1£(Q _^_ <^>(^) r SwtJ tnf(t)t-z zn zn+*' where ^> (^) denotes the integral, algebraical, polynomial V " f +0 j~ i -f ~ if +•••' when t is made zero. Hence and therefore which, by § GO, shews that/(V) is of class n. * Comptcs Rendus, t. xciv, (1882), p. G36. 92 CLASS-PROPERTIES OF [61. COROLLARY. The product of any finite number of functions of the same class n is a function of class not higher than n ; and the class of the product of any finite number of functions of different classes is not greater than the highest class of the component functions. The following are the chief references to memoirs discussing the class of functions : Laguerrc, Comptes Rendus, t. xciv, (1882), pp. 160-163, pp. 635—638, ib. t. xcv, (1882), pp. 828—831, ib. t. xcviii, (1884), pp. 79—81 ; Poincare, Bull, des Sciences Math., t. xi, (1883), pp. 136—144 ; Cesaro, Comptes Rendm, t. xcix, (1884), pp. 26—27, followed (p. 27) by a note by Hermite; Giornale di Battaglini, t. xxii, (1884), pp. 191 — 200; Vivanti, Giornale di Battaglini, t. xxii, (1884), pp. 243—261, pp. 378—380, ib. t. xxiii, (1885), pp. 96—122, ib. t. xxvi, (1888), pp. 303—314 ; Hermite, Cours d la faculte' des Sciences (4me ed., 1891), pp. 91 — 93. Ex. 1. The function 2 1=1 where the quantities c are constants, n is a finite integer, and the functions J\ (z) are algebraical polynomials, is of class unity. Ex. 2. If a simple function be of class %, its derivative is also of class n. Ex. 3. Discuss the conditions under which the sum of two functions, each of class n, is also of class n. Ex. 4. Examine the following test for the class of a function, due to Poincare. Let a be any number, no matter how small provided its argument be such that eaz vanishes when z tends towards infinity. Then / (z) is of class n, if the limit of vanish with indefinite increase of z. A possible value of a is 2 ciai~n~1, where C; is a constant of modulus unity. Ex. 5. Verify the following test for the class of a function, due to de Sparre*. Let X be any positive non-infinitesimal quantity ; then the function / (z) is of class n, if the limit, for m = oo , of \amn~l{\am + i\-\am\} be not less than X. Thus sin z is of class unity. Ex. 6. Let the roots of 0n + 1 = l be 1, a, a2, ...... , an; and let f (s) be a function of class n. Then forming the product n/(a«4 we evidently have an integral function of zn + 1; let it be denoted by F(zn + 1). The roots of * Comptes Rendus, t. cii, (1886), p. 741. 61.] TRANSCENDENTAL INTEGRAL FUNCTIONS 93 F(zn+l) = Q are a^'for i=l, 2, and s = 0, 1, , n\ and therefore, replacing zn + 1 by z, the roots ofF(z) = 0 are a?*1 for i=l, 2, ....... Since/ (z) is of class n, the series converges uniformly and unconditionally. This series is the sum of the first powers of the reciprocals of the roots of F(z}~ 0; hence, according to the definition (p. 89), F(z) is of class zero. It therefore follows that from, a function of any class a function of class zero with a modified variable can be deduced. Conversely, by appropriately modifying the variable of a given function of class zero, it is possible to deduce functions of any required class. Ex. 7. If all the zeros of the function =1 r anr \ be real, then all the zeros of its derivative are also real. (Witting.) 00 I / ~ \ U\(l--)e' «=* ^\ «W CHAPTER VI. FUNCTIONS WITH A LIMITED NUMBER OF ESSENTIAL SINGULARITIES. 62. SOME indications regarding the character of a function at an essential singularity have already been given. Thus, though the function is regular in the vicinity of such a point a, it may, like sn - at the origin, % have a zero of unlimited multiplicity or an infinity of unlimited multiplicity at the point ; and in either case the point is such that there is no factor of the form (z — a)x which can be associated with the function so as to make the point an ordinary point for the modified function. Moreover, even when the path of approach to the essential singularity is specified, the value acquired is not definite : thus, as z approaches the origin along the axis of x, so that its value may be taken to be 1 -f- (4>mK + x), the value of sn - is not z definite in the limit when m is made infinite. One characteristic of the point is the indefiniteness of value of the function there, though in the vicinity the function is uniform. A brief statement and a proof of this characteristic were given in § 33 ; the theorem there proved — that a uniform analytical function can assume any value at an essential singularity — may also be proved as follows. The essential singularity will be taken at infinity — a supposition that will be found not to detract from generality. Let f(z) be a function having any number of zeros and any number of accidental singularities and £ = oo for its sole essential singularity ; then it can be expressed in the form /w-88*"' where G1 (z) is algebraical or transcendental according as the number of zeros is finite or infinite and G2(z) is algebraical or transcendental according as the number of accidental singularities is finite or infinite. If Cr2 (z) be transcendental, we can omit the generalising factor e°(z). Then f(z) has an infinite number of accidental singularities ; each of them in the finite part of the plane is of only finite multiplicity and therefore some of them must be at infinity. At each such point, the function G2 (z) vanishes and Ol (z) does not vanish ; and so f(z) has infinite values for z = oo . 62.] VALUE AT AN ESSENTIAL SINGULARITY 95 If Gz (2) be algebraical and Gl (z) be also algebraical, then the factor ea(z) may not be omitted, for its omission would make f(z) an algebraical function. Now z = oo is either an ordinary point or an accidental singularity of ft <*)/<?.<*); hence as g (z} is integral there are infinite values of z which make infinite. If G.>.(z) be algebraical and G^ (z) be transcendental, the factor eg(z) maybe omitted. Let al5 a2,..., an be the roots of G2(z): then taking f(z)= ^- we have Ar= a non-vanishing constant ; and so where Gn (z) is a transcendental integral function. When 2 = oo , the value of G3(z)/G.,(z) is zero, but Gn(z) is infinite ; hence f(z) has infinite values for Z= 00 . Similarly it may be shewn, as follows, that/(z) has zero values for 0 = oo . In the first of the preceding cases, if Gl (z) be transcendental, so that f (z) has an infinite number of zeros, then some of them must be at an infinite distance; f(z) has a zero value for each such point. And if GI(Z) be algebraical, then there are infinite values of z which, not being zeros of G2(z), make f(z) vanish. In the second case, when z is made infinite with such an argument as to make the highest term in g(z) a real negative quantity, then f(z) vanishes for that infinite value of z. In the third case,/(V) vanishes for a zero of G1(z) that is at infinity. Hence the value of f (z) for z= oo is not definite. If, moreover, there be any value neither zero nor infinity, say G, which f(z) cannot acquire for z = oo , then /(*)-C is a function which cannot be zero at infinity and therefore all its zeros are in the finite part of the plane : no one of them is an essential singularity, for f(z) has only a single value at any point in the finite part of the plane; hence they are finite in number and are isolated points. Let H1 (z) be the alge braical polynomial having them for its zeros. The accidental singularities of f(z} — C are the accidental singularities of f(z) ; hence 96 FORM OF A FUNCTION NEAR [62. where, if G2(z) be algebraical, the exponential h(z) must occur, since f(z), and therefore f(z) — C, is transcendental. The function -| sy / \ TJ1 ( ~\ — _ 2 \ / 0—h (z) •* \*/ f t ~\ n TT f n\ J(z)-L H1 (z) evidently has z= oo for an essential singularity, so that, by the second or the third case above, it certainly has an infinite value for z = co , that is, f(z) certainly acquires the value G for z= GO . Hence the function can acquire any value at an essential singularity. 63. We now proceed to obtain the character of the expression of a function at a point z which, lying in the region of continuity, is in the vicinity of an essential singularity b in the finite part of the plane. With b as centre describe two circles, so that their circumferences and the whole area between them lie entirely within the region of continuity. The radius of the inner circle is to be as small as possible consistent with this condition; and therefore, as it will be assumed that b is the only singularity in its own immediate vicinity, this radius may be made very small. The ordinary point z of the function may be taken as lying within the circular ring-formed part of the region of continuity. At all such points in this band, the function is holomorphic ; and therefore, by Laurent's Theorem (§ 28), it can be expanded in a converging series of positive and negative integral powers of z — b in the form + V-L(Z — 6)"1 + v2 (z — 6)~2 + . . ., where the coefficients un are determined by the equation un = the integrals being taken positively round the outer circle, and the coefficients vn are determined by the equation the integrals being taken positively round the inner circle. The series of positive powers converges everywhere within the outer circle of centre b, and so (§ 26) it may be denoted by P (z - b) ; and the function P may be either algebraical or transcendental. The series of negative powers converges everywhere without the inner circle of centre b ; and, since 6 is not an accidental but an essential singularity of the function, the series of negative powers contains an infinite number of 63.] AN ESSENTIAL SINGULARITY 97 terms. It may be denoted by G I -- rh a series converging for all points \z — o/ in the plane except z = b and vanishing when z — b = co. Thus is the analytical representation of the function in the vicinity of its essential singularity b ; the function G is transcendental and converges everywhere in tlie plane except at z =• b, and the function P, if transcendental, converges uniformly and unconditionally for sufficiently small values of | z — b \ . Had the singularity at b been accidental, the function G would have been algebraical. COROLLARY I. If the function have any essential singularity other than b, it is an essential singularity of P (z — b) continued outside the outer circle ; but it is not an essential singularity of G ( -- j] , for the latter function \z — ol converges everywhere in the plane outside the inner circle. COROLLARY II. Suppose the function has no singularity in the plane except at the point b ; then the outer circle can have its radius made infinite. In that case, all positive powers except the constant term w0 disappear: and even this term survives only in case the function have a finite value at infinity. The expression for the function is and the transcendental series converges everywhere outside the infinitesimal circle round b, that is, everywhere in the plane except at the point b. Hence the function can be represented by This special result is deduced by Weierstrass from the earlier investiga tions*, as follows. If f(z) be such a function with an essential singularity at b, and if we change the independent variable by the relation Z/==^b' thcn/(V) changes into a function of z', the only essential singularity of which is at / = GO . It has no other singularity in the plane ; and the form of the function is therefore G (z'), that is, a function having an essential singularity at b but no other singularity in the plane is * Weierstrass (I.e.), p. 27. F. 98 FORM OF A FUNCTION NEAR [63. COROLLARY III. The most general expression of a function having its sole essential singularity at b a point in the finite part of the plane and any number of accidental singularities is G, where the zeros of the function are the zeros of GI, the accidental singularities of the function are the zeros of G2, and the function g in the exponential is a function which is finite everyiuhere except at b. This can be derived in the same way as before ; or it can be deduced from the corresponding theorem relating to transcendental integral functions, as above. It would be necessary to construct an integral function G2(z') having as its zeros and then to replace z by - — j ; and G., is algebraical or transcendental, Z 0 according as the number of zeros is finite or infinite. Similarly we obtain the following result : COROLLARY IV. A uniform function of z, which has its sole essential singularity at b a point in the finite part of the plane and no accidental singularities, can be represented in the form of an infinite product of primary factors of the form \z — b which converges uniformly and unconditionally everywhere in the plane except at z = b. The function g ( =] is an integral function of T vanishing when J \z-bj z-b r vanishes; and k and I are constants. In particular factors, q( T) z - b ^ \z - b) may vanish ; and either k or I (but not both k and I) may vanish with or without a vanishing exponent q { T ) . J \z-bj If tt{ be any zero, the corresponding primary factor may evidently be expressed in the form (z — ,z — Similarly, for a uniform function of z with its sole essential singularity at b and any number of accidental singularities, the product-form is at once derivable 63.] AN ESSENTIAL SINGULARITY 99 by applying the result of the present Corollary to the result given in Corollary III. These results, combined with the results of Chapter V., give the complete general theory of uniform functions with only one essential singularity. 64. We now proceed to the consideration of functions, which have a limited number of assigned essential singularities. The theorem of § 63 gives an expression for the function at any point in the band between the two circles there drawn. Let c be such a point, which is thus an ordinary point for the function ; then in the domain of c, the function is expansible in a form Pl (z — c). This domain may extend to an essential singularity b, or it may be limited by a pole d which is nearer to c than b is, or it may be limited by an essential singularity / which is nearer to c than b is. In the first case, we form a continuation of the function in a direction away from b; in the second case, we continue the function by associating with the function a factor (z — d)n which takes account of the accidental singularity ; in the third case, we form a continuation of the function towards f. Taking the continuations for successive domains of points in the vicinity of/, we can obtain the value of the function for points on two circles that have / for their common centre. Using these values, as in § 63, to obtain coefficients, we ultimately construct a series of positive and negative powers converging except at / for the vicinity of/ Different expressions in different parts of the plane will thus be obtained, each being valid only in a particular portion: the aggregate of all of them is the analytical expression of the function for the whole of the region of the plane where the function exists. We thus have one mode of representation of the function ; its chief advantage is that it indicates the form in the vicinity of any point, though it gives no suggestion of the possible modification of character elsewhere. This deficiency renders the representation insufficiently precise and complete ; and it is therefore necessary to have another mode of representation. 65. Suppose that the function has n essential singularities a,!, a*,..., an and that it has no other singularity. Let a circle, or any simple closed curve, be drawn enclosing them all, every point of the boundary as well as the included area (with the exception of the n singularities) lying in the region of continuity of the function. Let z be any ordinary point in the interior of the circle or curve ; and consider the integral , f/+\ I '*=-***> taken round the curve. If we surround z and each of the n singularities by small circles with the respective points for centres, then the integral round 7—2 100 FUNCTIONS WITH A LIMITED NUMBER [65. the outer curve is equal to the sum of the values of the integral taken round the n + l circles. Thus and therefore The left-hand side of the equation isf(z). Evaluating the integrals, we have where Gr is, as before, a transcendental function of - - — vanishing when 1 is zero. z — ar Now, of these functions, Gr{- -] converges everywhere in the plane \& \jbfj except at ar : and therefore, as n is finite, r=i \z - a is a function which converges everywhere in the plane except at the n points Clj , . . . , an . Because z = oc is not an essential singularity of f(z), the radius of the circle in the integral =—. . ! /-- dt may be indefinitely increased. The value ZTTI J s t — z of f(t) tends, with unlimited increase of t, to some determinate value G which is not infinite ; hence, as in § 24, II., Corollary, the value of the integral is C. We therefore have the result that/0) can be expressed in the form \z-a, or, absorbing the constant C into the functions G and replacing the limitation, that the function Gr(— — } shall vanish for — = 0, by the limitation \z — arj z — ar that, for the same value =0, it shall be finite, we have the theorem*:— z — ar If a given function f(z) have n singularities a^,..., an, all of which are in the finite part of the plane and are essential singularities, it can be expressed in the form 2G f-M, r=i r \z - aj ' * The method of proof, by an integration, is used for brevity : the theorem can be established by purely algebraical reasoning. 65.] OF ESSENTIAL SINGULARITIES 101 where Gr is a transcendental function converging everywhere in the plane except at ar and having a determinate finite value gr for - — = 0, such z — cir n that 2 gr is the finite value of the given function at infinity. r=l COROLLARY. If the given function have a singularity at oo , and n singu larities in the finite part of the plane, then the function can be expressed in the form w / 1 \ G(z) + SG,(— L-J, r=i \z-arr where Gr is a transcendental or an algebraic polynomial function, according as ar is an essential or an accidental singularity : and so also for G (z), accord ing to the character of the singularity at infinity. 66. Any uniform function, which has an essential singularity at z = a, can (§ 63) be expressed in the form for points z in the vicinity of a. Suppose that, for points in this vicinity, the function f(z) has no zero ; that it has no accidental singularity ; and therefore, among such points z, the function 1 df(z) /(*) dz has no pole, and therefore no singularity except that at a which is essential. Hence it can be expanded in the form G(^+P(z-a\ z-a where G converges everywhere in the plane except at a, and vanishes for = 0. Let z — a dz , / 1 \ where 0^ I ^— ^ I converges everywhere in the plane except at a, and vanishes for — — = o. z — a Then c, evidently not an infinite quantity, is an integer. To prove this, describe a small circle of radius p round a : then taking z-a = pe91 so that — = idd, we have z — a l M(*\ dz = P (z — a) dz 102 FUNCTIONS WITH A LIMITED NUMBER [66. and therefore Now JP(z — a)dz is a uniform function : and so is f(z). But a change of 6 into 6 + 2-7T does not alter z or any of the functions : thus actotr — 1 • ~~ *• i and therefore c is an integer. 67, If the function /(z) have essential singularities alt..., an and no others, then' it can be expressed in the form n /I C+ $9J-± r=i \z-ar If there be no zeros for this function f(z) anywhere (except of course such as may enter through the indeterminateness at the essential singularities), then /(*) dz has n essential singularities a1}..., an and no other singularities of any kind. Hence it can be expressed in the form n / 1 \ C+ 2 Gr(- -), r=i \z-a,rl where the function Gr vanishes with . Let z — ar cr d I T~~ \js — a,./ z — ar dz { r\z — ar where Gr I ) is a function of the same kind as Gr ( ) . \z — ar/ \z — arj Then all the coefficients cr, evidently not infinite quantities, are integers. For, let a small circle of radius p be drawn round ar : then, if z — ar = peei, we have crdz z — ar = cri6, and — ^ — = dPs (z - ar). z — as We proceed as before : the expression for the function in the former case is changed so that now the sum 2Pg(0— ar) for 5 = !,..., i — 1, r + 1,..., n is a uniform function; there is no other change. In exactly the same way as before, we shew that every one of the coefficients cr is an integer. Hence it appears that if a given function f(z) have, in the finite part of 67.] OF ESSENTIAL SINGULARITIES 103 the plane, n essential singularities al,..., an and no other singularities and if it have no zeros anywhere in the plane, then f(z) dz where all the coefficients c* are integers, and the functions G converge every where in the plane except at the essential singularities and Gi vanishes for -J-- 0. Now, since f(z) has no singularity at oo , we have for very large values of z and /'W = _>_ Z* and therefore, for very large values of z, _ f(z) dz u0 z2 z3 Thus there is no constant term in =7-^ ^r-^ , and there is no term in -. But /(*) dz z the above expression for it gives G as the constant term, which must therefore vanish ; and it gives 2c; as the coefficient of - , for -7- •< (r< ( - — H will begin z dz [ \z — ftj/ J with — at least ; thus ^a must therefore also vanish. Z" Hence for a function f (z) which has no singularity at z= oo and no zeros anywhere in the plane and of which the only singularities are the n essential singularities at a1} a2,..., an, we have / (z) dz i=i z - Oi i=i dz ( \z- a where the coefficients a are integers subject to the condition n 2 ct = 0. i=l If an= oo , so that 2= GO is an essential singularity in addition to a2, a2,..., an_j, there is a term 6r (z) instead of Gn( — - ] ; there is no term, that corre- \Z — C^n/ /^ spends to - — , but there may be a constant G. Writing — z — with the condition that G (z) vanishes when z — 0, we then have - = __ g ^ i=iz-at dz( v /J ». 104 PRODUCT-EXPRESSION OF [67. where the coefficients d are integers, but are no longer subject to the condition that their sum vanishes. Let R* (z} denote the function the product extending over the factors associated with the essential sin gularities of f(z) that lie in the finite part of the plane; thus R*(z) is a rational algebraical rneromorphic function. Since 1 dR*(z) = 2 d R* (z) dz ~ i=\z — a,i' we have 1 df(z) _ 1 dR*(z) =$ d_(-Q ( * f(z) dz R* (z) dz i=\dz\ l\z — a^ where Gn ( — - ) is to be replaced by G (z) if an = <x> , that is, if z — oo be an \z — anj essential singularity off(z). Hence, except as to an undetermined constant factor, we have t=i which is therefore an analytical representation of a function with n essential singularities, no accidental singularities, and no zeros: and the rational alge braical function R* (z) becomes zero or oo only at the singularities off(z). If z = oo be not an essential singularity, then R* (z) for z = oo is equal to M unity because 2 Cf = 0. 1=1 COROLLARY. It is easy to see, from § 43, that, if the point a; be only an accidental singularity, then a is a negative integer and wj I — - ) is zero : so \Z — Oii/ that the polar property at c^ is determined by the occurrence of a factor (z — a{)Ci solely in the denominator of the rational meromorphic function R* (z). And, in general, each of the integral coefficients a is determined from the expansion of the function f'(z) +f(z) in the vicinity of the singularity with which it is associated. 68. Another form of expression for the function can be obtained from the preceding; and it is valid even when the function has zeros not absorbed into the essential singularities f. Consider a function with one essential singularity, and let a be the point ; and suppose that, within a finite circle of centre a (or within a finite simple curve which encloses a), there are m simple zeros a, /3,..., X of the + See Guichard, TMorie des points singuliers essentiels, (These, Gauthier-Villars, Paris, 1883), especially the first part. 68.] A FUNCTION 105 function f(z) — m being assumed to be finite, and it being also assumed that there are no accidental singularities within the circle. Then, if /(*) = (* - «) (z - /3). . .(* -\)F (z\ the function F (z) has a for an essential singularity and has no zeros within the circle. Hence, for points z within the circle, where (?, ( ----- ) converges everywhere in the plane and vanishes with - — , \z — a] z a and P(z — a) is an integral function converging uniformly and unconditionally within the circle ; moreover, c is an integer. Thus F(z) = A(z- a)" eGl Let (*-a)(*-/3)...(*-X) = (*-a _ ( y _ r, \m a) then f(z) = (* - dTffl -- F(z} \z u/ = A(z- ar+°gi (~}eG> ^ e ^~a] "z . \z — ft/ Now of this product- expression for/(V) it should be noted: — (i) That m + c is an integer, finite because m and c are finite : <?,—- (ii) The function e ' ^z~a' can be expressed in the form of a series con verging uniformly and unconditionally everywhere, except at z = a, and proceeding in powers of - — in the form z a .... z — a (z — af It has no zero within the circle considered, for F (z) has no zero. Also gl(- - 1 \z a/ algebraical function of — ' — , beginning with unity and containing only is an z — a a finite number of terms : hence, multiplying the two series together, we have as the product a series proceeding in powers of - in the form £ ~" a — a which converges uniformly and unconditionally everywhere outside any small circle round a, that is, everywhere except at a. Let this series be denoted by 106 PRODUCT-EXPRESSION OF [68. H I ]; it has an essential singularity at a and its only zeros are the \z-aj points a, (3,..., X, for the series multiplied by gl (— -) has \z — ft/ no zeros : (iii) The function fP (z — a) dz is a series of positive powers of z — a, converging uniformly in the vicinity of a; and therefore Q^(z-d)dz can ke expanded in a series of positive integral powers of z — a which converges in the vicinity of a. Let it be denoted by Q (z — a) which, since it is a factor of F (z), has no zeros within the circle. Hence we have /(*) = A (z - aYQ (z - a) H where p, is an integer ; H ( — - J is a series that converges everywhere except at a, is equal to unity when - vanishes, and has as its zeros the (finite) z — a number of zeros assigned to f(z) within a finite circle of centre a ; and Q (z — a) is a series of positive powers of z — a beginning with unity which converges — but has no zero — within the circle. The foregoing function f(z) is supposed to have no essential singularity except at ft. If, however, a given function have singularities at points other than a, then the circle would be taken of radius less than the distance of a from the nearest essential singularity. Introducing a new function f{ (z} defined by the equation the value of /[ (z) is Q (z — a) within the circle, but it is not determined by the foregoing analysis for points without the circle. Moreover, as (z — a)* and also Hi— ] are finite everywhere except possibly at a, it follows that essential singularities of f(z) other than a must be essential singu larities of fj (z). Also since /i (z) is Q(z — a) in the immediate vicinity of a, this point is not an essential singularity of /i (z). Thus /i (z) is a function of the same kind as f(z) ; it has all the essential singularities of f(z) except ft, but it has fewer zeros, on account of the m zeros of f(z) possessed by H ( - — ] . The foregoing expression for f(z) is \Z — ft/ the one referred to at the beginning of the section. If we choose to absorb into /x (z} the factors e \z~a' and e?P(z~° dz, which occur in (z - $*• ffl f Jil ^ (T- a) \2 — ft/ 68.] A FUNCTION 107 an expression that is valid within the circle considered, then we obtain a result that is otherwise obvious, by taking where now g± (— — ) is algebraical and has for its zeros all the zeros within \Z — d/ the circle ; yu, is an integer; and/j (z) is a function of the same kind as f(z), which now possesses all the essential singularities of f(z} but has zeros fewer by the in zeros that are possessed by z— a 69. Next, consider a function f(z) with n essential singularities al} a2,..., an but without accidental singularities; and let it have any number of zeros. When the zeros are limited in number, they may be taken to be isolated points, distinct in position from the essential singularities. When the zeros are unlimited in number, then at least one of the singularities must be such that an infinite number of the zeros lie within a circle of finite radius, described round it as centre and containing no other singularity. For if there be not an infinite number in such a vicinity of some one point (which can only be an essential singularity, otherwise the function would be zero everywhere), then the points are isolated and there must be an infinite number at z = oo . If z = oo be an essential singularity, the above alternative is satisfied : if not, the function, being continuous save at singularities, must be zero at all other parts of the plane. Hence it follows that if a uniform function have a finite number of essential singularities and an infinite number of zeros, all but a finite number of the zeros lie within circles of finite radii described round the essential singularities as centres ; at least one of the circles contains an infinite number of the zeros, and some of the circles may contain only a finite number of them. We divide the whole plane into regions, each containing one but only one singularity and containing also the circle round the singularity ; let the region containing a{ be denoted by Ci, and let the region Gn be the part of the plane other than Glt (72, ...... , Gn_^. If the region G1 contain only a limited number of the zeros, then, by § 68, we can choose a new function /i (z) such that, if the function /j (z) has av for an ordinary point, has no zeros within the region Glt and has a2, a3, ...... , an for its essential singularities. If the region Cl contain an unlimited number of the zeros, then, as in Corollaries II. and III. of § 63, we construct any transcendental function 108 GENERAL FORM OF A FUNCTION [69. 5xf— — ) , having a^ for its sole essential singularity and the zeros in GI for \z — OiJ all its zeros. When we introduce a function g: (2), defined by the equation the function g:{z) has no zeros in GI and certainly has a2, a3, ...... , an for essential singularities ; in the absence of the generalising factor of Glt it can have Hi for an essential singularity. By § G7, the function ~g{ (2), defined by gi (z) = 0 - cOc' ehl ^W , has no zero and no accidental singularity, and it has a^ as its sole essential singularity : hence, properly choosing cx and hi, we may take ft(*)-?i(*)/i(*)« so that fi (z) does not have aj as an essential singularity, but it has all the remaining singularities of ^ (z), and it has no zeros within C^. In either case, we have a new function ft (z) given by where /^ is an integer, the zeros off(z) that lie in GI are the zeros of GI ; the function fi(z) has «2, »s> ...... > an (but not a^ for its essential singularities, and it has the zeros of f(z) in the remaining regions for its zeros. Similarly, considering (72, we obtain a function /2 (z), such that where /A.2 is an integer, G2 is a transcendental function finite everywhere except at a2 and has for its zeros all the zeros of ft (z) — and therefore all the zeros of f (z) — that lie in G2 ; then f.2 (z) possesses all the zeros of f(z) in the regions other than GI and C2, and has a3, a4,..., an for its essential singularities. Proceeding in this manner, we ultimately obtain a function fn (z) which has none of the zeros off(z) in any of the n regions GI, C2,..., Cn, that is, has no zeros in the plane, and it has no essential singularities ; it has no acci dental singularities, and therefore fn(z) is a constant. Hence, when we • substitute, and denote by S* (z} the product II (z — a^1, we have Z — as the most general form of a function having n essential singularities, no accidental singularities, and any number of zeros. The function S* (z) is a rational algebraical function of z, usually meromorphic inform, and it has the essential singularities off(z) as its zeros and poles ; and the zeros of f (z) are distributed among the functions Gt. As however the distribution of the zeros by the regions C and therefore 69.] WITH ESSENTIAL SINGULARITIES 109 the functions G[ ) are somewhat arbitrary, the above form though general \z — a] is not unique. If any one of the singularities, say am, had been accidental and not essential, then in the corresponding form the function Gm ( - - ) would be \Z — dm/ algebraical arid not transcendental. 70. A function f(z], which has any finite number of accidental singu larities in addition to n assigned essential singularities and any number of assigned zeros, can be constructed as follows. Let A (z) be the algebraical polynomial which has, for its zeros, the accidental singularities of f(z), each in its proper multiplicity. Then the product /(*)-A(*) is a function which has no accidental singularities ; its zeros and its essential singularities are the assigned zeros and the assigned essential singularities of f (z) and therefore it is included in the form n ( S*(z)U \0t i=i ( where S* (z) is a rational algebraical meromorphic function having the points Oi, a.,,..., an for zeros and poles. The form of the function f(z} is therefore A )}• -ail) 71. A function f (z), which has an unlimited number of accidental singu larities in addition to n assigned essential singularities and any number of assigned zeros, can be constructed as follows. Let the accidental singularities be of, /3',.... Construct a function f^ (z), having the n essential singularities assigned to f (z}, no accidental singu larities, and the series a!, /3',. . . of zeros. It will, by § 69, be of the form of a product of n transcendental functions Gn+1,..., G.2n which are such that a function G has for its zeros the zeros oif-i(z} lying within a region of the plane, divided as in § 69 ; and the function Gn+t is associated with the point at-. Thus / (z) = T*-(z) ft Gn f=i where T* (z) is a rational algebraical meromorphic function having its zeros and its poles, each of finite multiplicity, at the essential singularities ofy(^). Because the accidental singularities of f(z) are the same points and have the same multiplicity as the zeros of /i (z), the function / (z) /x (z) has no accidental singularities. This new function has all the zeros of f(z), and al,...,an are its essential singularities; moreover, it has no accidental singu larities. Hence the product f(z)fi (z) can be represented in the form 110 GENERAL FORM OF A FUNCTION [71. and therefore we have z-fi (f-a) as an expression of the function. But, as by their distribution through the n selected regions of the plane in § 69, the zeros can to some extent be arbitrarily associated with the functions Gl} G*,,..., Gn and likewise the accidental singularities can to some extent be arbitrarily associated with the functions Gn+l, Gn+»,..., G^i, the product-expression just obtained, though definite in character, is not unique in the detailed form of the functions which occur. S* (z) The fraction 7**) \ is algebraical and rational ; and it vanishes or becomes infinite only at the essential singularities alt a.2,..., an, being the product of factors of the form (z — «i)ms for i = l, 2,..., n. Let the power (z — a^ be absorbed into the function G{/Gn+i for each of the n values of i ; no substantial change in the transcendental character of Gi and of Gn+i is thereby caused, and we may therefore use the same symbol to denote the modified function after the absorption. Hence "f" the most general product-expression of a uniform function of z which has n essential singularities al} a*,..., an, any unlimited number of assigned zeros and any unlimited number of assigned accidental singularities is n ^ n — \z-an The resolution of a transcendental function with one essential singularity into its primary factors, each of which gives only a single zero of the function, has been obtained in § 63, Corollary IV. We therefore resolve each of the functions G^..., Gm into its primary factors. Each factor of the first n functions will contain one and only one zero of the original functions / (.z) ; and each factor of the second n functions will contain one and only one of the poles of f(z). The sole essential singularity of each primary factor is one of the essential singularities off(z). Hence we have a method of constructing a uniform function with any finite number of essential singularities as a uniformly converging product of any number of primary factors, each of which has one of the essential singularities as its sole essential singularity and either (i) has as its sole zero either one of the zeros t Weierstrass, I.e., p. 48. 71.] WITH ESSENTIAL SINGULARITIES 111 or one of the accidental singularities of/(V), so that it is of the form Z — € \ a ( — . or (ii) it has no zero and then it is of the form /fe). When all the primary factors of the latter form are combined, they constitute a generalising factor in exactly the same way as in § 52 and in § 63, Cor. III., except that now the number of essential singularities is not limited to unity. Two forms of expression of a function with a limited number of essential singularities have been obtained : one (§ 65) as a sum, the other (§ 69) as a product, of functions each of which has only one essential singularity. Inter mediate expressions, partly product and partly sum, can be derived, e.g. expressions of the form z— c. But the pure product-expression is the most general, in that it brings into evidence not merely the n essential singularities but also the zeros and the accidental singularities, whereas the expression as a sum tacitly requires that the function shall have no singularities other than the n which are essential. Note. The formation of the various elements, the aggregate of which is the complete representation of the function with a limited number of essential singularities, can be carried out in the same manner as in § 34 ; each element is associated with a particular domain, the range of the domain is limited by the nearest singularities, and the aggregate of the singularities forms the boundary of the region of continuity. To avoid the practical difficulty of the gradual formation of the region of continuity by the construction of the successive domains when there is a limited number of singularities (and also, if desirable to be considered, of branch-points), Fuchs devised a method which simplifies the process. The basis of the method is an appropriate change of the independent variable. The result of that change is to divide the plane of the modified variable f into two portions, one of which, G2, is finite in area and the other of which, Gl, occupies the rest of the plane; and the boundary, common to Gl and G2, is a circle of finite radius, called the discriminating circle* of the function. In G2 the modified function is holomorphic ; in G^ the function is holomorphic except at f = oo ; and all the singularities (and the branch-points, if any) lie on the discriminating circle. The theory is given in Fuchs's memoir " Ueber die Darstellung der Functionen com- plexer Variabeln, ," Crelle, t. Ixxv, (1872), pp. 176 — 223. It is corrected in details and is amplified in Crelle, t. cvi, (1890), pp. 1 — 4, and in Crelle, t. cviii, (1891), pp. 181—192; see also Nekrassoff, Math. Ann., t. xxxviii, (1891), pp. 82—90, and Anissimoff, Math. Ann., t. xl, (1892), pp. 145—148. * Fuchs calls it Grenzkreis. CHAPTER VII. FUNCTIONS WITH UNLIMITED ESSENTIAL SINGULARITIES, AND EXPANSION IN SERIES OF FUNCTIONS. 72. IT now remains to consider functions which have an infinite number of essential singularities*. It will, in the first place, be assumed that the essential singularities are isolated points, that is, that they do not form a continuous line, however short, and that they do not constitute a continuous area, however small, in the plane. Since their number is unlimited and their distance from one another is finite, there must be at least one point in the plane (it may be at z = oo ) where there is an infinite aggregate of such points. But no special note need be taken of this fact, for the character of an essential singularity has not yet entered into question ; the essential singu larity at such a point would merely be of a nature different from the essential singularity at some other point. We take, therefore, an infinite series of quantities a1} a.2, a3,... arranged in order of increasing moduli, and such that no two are the same : and so we have infinity as the limit of av when v = <x> . Let there be an associated series of uniform functions of z such that for all values of i. the function G'i ( ) , vanishing with , has a{ as its \Z - Of/ Z — Oi * The results in the present chapter are founded, except where other particular references are given, upon the researches of Mittag-Leffler and Weierstrass. The most important investigations of Mittag-Leffler are contained in a series of short notes, constituting the memoir " Sur la th6orie des fonctions uniformes d'une variable," Comptes Rendus, t. xciv, (1882), pp. 414, 511, 713, 781, 938, 1040, 1105, 1163, t. xcv, (1882), p. 335 ; and in a memoir " Sur la representation analytique des fonctions monogenes uniformes," Acta Math., t. iv, (1884), pp. 1 — 79. The investigations of Weierstrass referred to are contained in his two memoirs " Ueber einen functionentheoretischen Satz des Herrn G. Mittag-Leffler," (1880), and " Zur Functionenlehre," (1880), both included in the volume Abhandlungen aus der Functionenlehre, pp. 53 — 66, 67 — 101, 102 — 104. A memoir by Hermite " Sur quelques points de la theorie des fonctions," Acta Soc. Fenn., t. xii, pp. 67 — 94, Crelle, t. xci, (1881), pp. 54 — 78 may be consulted with great advantage. 72.] MITTAG-LEFFLER'S THEOREM 113 sole singularity; the singularity is essential or accidental according as GI is transcendental or algebraical. These functions can be constructed by theorems already proved. Then we have the theorem, due to Mittag- Le frier: — It is always possible to construct a uniform analytical function F (z), having no singularities other than a1} a«, a,, ... and such that for each deter minate value of v, the difference F (z)-Gv ( ) is finite for z = av and \z av/ therefore, in the vicinity of av, is expressible in the form P (z — «„). 73. To prove Mittag-Leffler's theorem, we first form subsidiary functions Fv (z), derived from the functions G as follows. The function Gv (—- — } \z — aj converges everywhere in the plane except at the point «„; hence within a circle z < av\ it is a monogenic analytic function of z, and can therefore be expanded in a series of positive powers of z which converges uniformly within the circle, say z-a for values of z such that \z\ < av . If a,, be zero, there is evidently no expansion. Let e be a positive quantity less than 1, and let elf e2, e3, ... be arbitrarily chosen positive decreasing quantities, subject to the single condition that 2e is a converging series, say of sum A : and let e0 be a positive quantity inter mediate between 1 and e. Let g be the greatest value of ~ f z — a, for points on or within the circumference \z\ = e0 a,|; then, because the series 00 2 v^z* is a converging series, we have, by § 29, or Hence, with values of z satisfying the condition \z\^.e av\, we have, for any value of m, /j.=m Vu Z 2, q - 9 mJt n = m fco 1- since e<e0. Take the smallest integral value of m such that 9 F. 114 MITTAG-LEFFLER'S it will be finite and may be denoted by mv : and thus we have [73. for values of z satisfying the condition \z\^.e av\. We now construct a subsidiary function Fv (z) such that, for all values of z, then for values of UL which are ^ e aJ, Moreover, the function 2 zv^ is finite for all finite values of z so that, if we n=o take .j -a — i then 6,,(^) is zero at infinity, because, when 5=00, #„(- -)is finite by \z — civ/ hypothesis. Evidently <f>v(z) is infinite only at z = av, and its singularity is of the same kind as that of Gt z — a, 74. Now let c be any point in the plane, which is not one of the points «], a2, as, ...; it is possible to choose a positive quantity p such that no one of the points a is included within the circle z — c = p- Let av be the singularity, which is the point nearest to the origin satisfy ing the condition «„ > c \ + p ; then, for points within or on the circle, we have ' z as when s has the values v, v + 1, v + 2, Introducing the subsidiary functions Fv (z), we have, for such values of z, and therefore F.(z) a finite quantity. It therefore follows that the series 2 F, (z) converges uni- 8=v formly and unconditionally for all values of z which satisfy the condition 74.] THEOREM 115 z — c\^.p. Moreover, all the functions Fl(z), F2(z), ..., Fr_l(z] are finite for such values of z, because their singularities lie without the circle z — c = p ; and therefore the series S Fr(z) r=l converges uniformly and unconditionally for all points z within or on the circle \z — c =p, where p is chosen so that the circle encloses none of the points a. The function, represented by the series, can therefore be expanded in the form P (z — c), in the domain of the point c. If am denote any one of the points a1} a2, ..., and we take p' so small that all the points, other than am, lie without the circle I / I * U"m — P ) then, since Fm (z) is the only one of the functions F which has a singularity at am, the series ^{Fr(z}}-Fm(z) converges regularly in the vicinity of a, and therefore it can be expressed in the form P (z — am). Hence a the difference of Fm and Gm being absorbed into the series P to make Pj. It GO thus appears that the series 2 Fr (z) is a function which has infinities only r = \ at the points a1} a2, ..., and is such that can be expressed in the vicinity of am in the form P (z - am). Hence 2 Fr (z) is a function of the required kind. 75. It may be remarked that the function is by no means unique. As the positive quantities e were subjected to merely the single condition that they form a converging series, there is the possibility of wide variation in their choice: and a difference of choice might easily lead to a difference in the ultimate expression of the function. This latitude of ultimate expression is not, however, entirely unlimited. For, suppose there are two functions F(z) and F (z\ enjoying all the assigned properties. Then as any point c, other than a^, a2, ..., is an ordinary point for both F (z) and F (z), it is an ordinary point for their difference : and so F(z)-F(z) = P(z-c) 8—2 116 FUNCTIONS POSSESSING [75. for points in the immediate vicinity of c. The points a are, however, singularities for each of the functions : in the vicinity of such a point a* we have since the functions are of the required form : hence F(z}-F(z}=P(z-ai) -P(z- ai), or the point a; is an ordinary point for the difference of the functions. Hence every finite point in the plane, whether an ordinary point or a singularity for each of the functions, is an ordinary point for the difference of the functions : and therefore that difference is a uniform integral function of z. It thus appears that, if F (z) be a function with the required properties, then every other function with those properties is of the form F(z) + G(z], where G (z) is a uniform integral function of z either transcendental or algebraical. The converse of this theorem is also true. 00 Moreover, the function G (z) can always be expressed in a form 2 gv(z), if v=\ it be desirable to do so : and therefore it follows that any function with the assigned characteristics can be expressed in the form 76. The following applications, due to Weierstrass, can be made so as to give a new expression for functions, already considered in Chapter VI., having z = oo as their sole essential singularity and an unlimited number of poles at points Oi, a2, — If the pole at af be of multiplicity mi} then (z — a$n>f(z) is regular at the point a; and can therefore be expressed in the form mi— 1 Hence, if we take /f (z) = 2 c^ (z — ai)~TO<+'t, M = 0 we have f(z} =fi (z) + P (z — «;). Now deduce from fi(z) a function Fi(z) as in | 73, and let this deduction be effected for each of the functions /,- (z). Then we know that is a uniform function of z having the points a1} a2, ... for poles in the proper 76.] UNLIMITED SINGULARITIES 117 multiplicity and no essential singularity except z = oo . The most general form of the function therefore is r=\ Hence any uniform analytical function which has no essential singularity except at infinity can be expressed as a sum of functions each of which has only one singularity in the finite part of the plane. The form of Fr (z) is fr(z}-Gr(z\ where fr (z) is infinite at z = ar and Gr (z) is a properly chosen integral function. We pass to the case of a function having a single essential singularity at c and at no other point and any number of accidental singularities, by taking z' = - as in § 63. Cor. II.: and so we obtain the theorem : z — c Any uniform function which has only one essential singularity, which is at c, can be expressed as a sum of uniform functions each of which has only one singularity different from c. Evidently the typical summative function Fr (z) for the present case is of the form Z — 77. The results, which have been obtained for functions possessed of an infinitude of singularities, are valid on the supposition, stated in § 72, that the limit of av with indefinite increase of v is infinite ; the series ttj, «2, ••• tends to one definite limiting point which is 2=00 and, by the substitution z' (z — c) = 1, can be made any point c in the finite part of the plane. Such a series, however, does not necessarily tend to one definite limiting point: it may, for instance, tend to condensation on a curve, though the condensation does not imply that all points of the continuous arc of the curve must be included in the series. We shall not enter into the discussion of the most general case, but shall consider that case in which the series of moduli \al) a2 , ... tends to one definite limiting value so that, with in definite increase of v, the limit of \av is finite and equal to R ; the points «i, «2, ... tend to condense on the circle \z = R. Such a series is given by 2fori ( I _ l -m+n «„,*={! + for &=0, 1, ..., n, and n=l, 2, ... ad inf.; and another* by a«Hl + (-l)ncn}e2M7"V2, where c is a positive proper fraction. * The first of these examples is given by Mittag-Leffler, Acta Math., t. iv, p. 11 ; the second was stated to me by Mr Burnside. 118 FUNCTIONS POSSESSING [77. With each point am we associate the point on the circumference of the circle, say bm, to which am is nearest: let | dm "m I = Pm> so that pm approaches the limit zero with indefinite increase of m. There cannot be an infinitude of points ap, such that pp^<&, any assigned positive quantity ; for then either there would be an infinitude of points a within or on the circle \z\ = R — ®, or there would be an infinitude of points a within or on the circle z = R + ©, both of which are contrary to the hypothesis that, with indefinite increase of v, the limit of \av is R. Hence it follows that a finite integer n exists for every assigned positive quantity ®, such that \am-bm\ < ® when m^n. Then the theorem, which corresponds to Mittag-LefHer's as stated in § 72 and which also is due to him, is as follows : — It is always possible to construct a uniform analytical function of z which exists over the whole plane, except at the points a and b, and which, in the immediate vicinity of each one of the singularities a, can be expressed in the form where the functions G{ are assigned functions, vanishing with - - and finite Z — (Li everywhere in the plane except at the single points a; with which they are respectively associated. In establishing this theorem, we shall need a positive quantity e less than unity and a converging series e^ e2, e3, ... of positive quantities, all less than unity. Let the expression of the function Gn be "I / _. .. \0 I / - _. \5 ' ' ' ' ' n \z - a ~ z-an (z- an)2 (z - an)s Then, since z - an = (z - bn) \l -- — ~l\ , ( z on ) the function Gn can be expressed* in the form l«— <li for values of z such that an - z-bn and the coefficients A are given by the equations * The justification of this statement is to be found in the proposition in § 82. 77.] UNLIMITED SINGULARITIES 119 Now, because Gn is finite everywhere in the plane except at an, the series has a finite value, say #, for any non-zero value of the positive quantity %n ; then Hence 0*-!)! ft & f < S flfr-^ 71 ?^ Introducing a positive quantity a such that we choose £n so that £n < a|an - bn\ ; and then | A n> ^ \ < go. ( 1 + a)*-1. Because (1 + a) e is less than unity, a quantity 6 exists such that (1 + a) e < 6 < 1. Then for values of z determined by the condition go. 6 dn on < e, we have al-0' Let the integer mn be chosen so that ga &> it will be a finite integer, because 0< 1. Then 00 (1 7) V I A I "H ^^ We now construct, as in § 73, a subsidiary function Fn(z), defining it by the equation so that for points z determined by the condition \Fn(z)\<en. A function with the required properties is 00 Fm(z\ < €, we have m=l 120 FUNCTIONS POSSESSING [77. To prove it, let c be any point in the plane distinct from any of the points a and b ; we can always find a value of p such that the circle \z-c\=p contains none of the points a and b. Let I be the shortest distance between this circle and the circle of radius R, on which all the points b lie ; then for all points z within or on the circle z — c — p we have Now we have seen that, for any assigned positive quantity <s), there is a finite integer n such that I dm — bm < © when m ^ n. Taking ® = el, we have m < e when m^n,n being the finite integer associated with the positive quantity el. It therefore follows that, for points z within or on the circle \z — c\ = p, \Fm(z}\<em, when m is not less than the finite integer n. Hence a finite quantity because e1} e2, ... is a converging series; and therefore is a converging series. Each of the functions F1(z), F»(z), ..., Fn_-i(z) is finite when z — c ^ p ; and therefore is a series which converges uniformly and unconditionally for all values of z included in the region \z-c\^p. Hence the function represented by the series can be expressed in the form P (z — c) for all such values of z. The function therefore exists over the whole plane except at the points a and b. It may be proved, exactly as in § 74, that, for points z in the immediate vicinity of a singularity am, The theorem is thus completely established. The function thus obtained is not unique, for a wide variation of choice of the converging series ea + e2 + . . . is possible. But, in the same way as in the 77.] UNLIMITED SINGULARITIES 121 corresponding case in § 75, it is proved that, if F (z) be a function with the required properties, every other function with those properties is of the form F(z}+G(z\ where G (z) behaves regularly in the immediate vicinity of every point in the plane except the points b. 78. The theorem just given regards the function in the light of an infinite converging series of functions of the variable : it is natural to suppose that a corresponding theorem holds when the function is expressed as an infinite converging product. With the same series of singularities as in § 77, when the limit of av with indefinite increase of v is finite and equal to R, the theorem* is: — It is always possible to construct a uniform analytical function which behaves regularly everywhere in the plane except at the points a and b and which in the vicinity of any one of the points av can be expressed in the form where the numbers w1} n2, ... are any assigned integers. The proof is similar in details to proofs of other propositions and it will therefore be given only in outline. We have au- provided such values of z, z-av z-bv z - bv ^i V z - bv J ' < e, the notation being the same as in § 77. Hence, for =e (/7 _ 7) \ i_ ^ _ M 2-bJ -n,, S by Ev (z), we have Ev (z} =e m" Hence, if F(z) denote the infinite product we have F(z) = e and F(z) is a determinate function provided the double series in the index of the exponential converge. * Mittag-Leffler, Acta Math., t. iv, p. 32 ; it may be compared with Weierstrass's theorem in §67. 122 TRANSCENDENTAL FUNCTION AS Because nv is a finite integer and because [78. is a converging series, it is possible to choose an integer mv so that 7) "x M(T^ where t]v is any assigned positive quantity. We take a converging series of positive quantities rjv : and then the moduli of the terms in the double series form a converging series. The double series itself therefore converges uniformly and unconditionally ; and then the infinite product F (z) converges uniformly and unconditionally for points z such that &„ — b.. < e. As in § 77, let c be any point in the plane, distinct from any of the points a and b. We take a finite value of p such that the circle z — c\=p contains none of the points a and b ; and then, for all points within or on this circle, z— <e when m^n, n being the finite integer associated with the positive quantity el. The product fi Ev(z) v=n is therefore finite, for its modulus is less than CO S IJK K = » the product n v=l is finite, because the circle z — c\ = p contains none of the points a and 6; and therefore the function F(z) is finite for all points within or on the circle. Hence in the vicinity of c, the function can be expanded in the form P (z — c) ; and therefore the function exists everywhere in the plane except at the points a and b. The infinite product converges ; it can be zero only at points which make one of the factors zero and, from the form of the factors, this can take place only at the points av with positive integers nv. In the vicinity of av all the factors of F (z) except Ev (z) are regular ; hence F (z)\Ev (z) can be expressed as a function of z — av in the vicinity. But the function has no zeros there, and therefore the form of the function is Pl (z-a,,). 78.] AN INFINITE SERIES OF FUNCTIONS 123 Hence in the vicinity of av, we have on combining with Pl (z — av) the exponential index in Ev(z). This is the required property. Other general theorems will be found in Mittag-Leffler's memoir just quoted. 79. The investigations in §§ 72 — 75 have led to the construction of a function with assigned properties. It is important to be able to change, into the chosen form, the expression of a given function, having an infinite series of singularities tending to a definite limiting point, say to z = oo . It is necessary for this purpose to determine (i) the functions Fr(z) so that the 00 series 2 Fr (z) may converge uniformly and (ii) the function G (z). r=l Let <& (z) be the given function, and let S be a simple contour embracing the origin and /j, of the singularities, viz., al , ...... , aM: then, if t be any point, we have - « « . m r« *£) ,,y r« *y) ,,. J t-z\t) J t-z\t) f(a) _ where I implies an integral taken round a very small circle centre a. If the origin be one of the points a1} a2, ...... , then the first term will be included in the summation. Assuming that z is neither the origin nor any one of the points a1} ..., a^, we have so 27TI AT ^ Now — . 7-^-7 dt 1 [(0)$>(t)fz\ — . 7-^-7 Ziri] t-z\tj , — -—. 2 I 7—^- - dt. t-Z\t) (ffl-l)i I \~dm 1(®(t) + ^i^+^ [ 124 TRANSCENDENTAL FUNCTION AS [79. unless z = 0 be a singularity and then there will be no term G (z). Similarly, it can be shewn that / I \ m-l / z \ A is equal to Gv(- -} - 2 vj-} = F, (z), \z - aj A=0 \aj where , — s— • 2?rt and the subtractive sum of m terms is the sum of the first m terms in the development of Gv in ascending powers of z. Hence If, for an infinitely large contour, m can be chosen so that the integral t- diminishes indefinitely with increasing contours enclosing successive singu larities, then The integer m may be called the critical integer. If the origin be a singularity, we take and there is then no term G (z) : hence, including the origin in the summa tion, we then have so that if, for this case also, there be some finite value of m which makes the integral vanish, then Other expressions can be obtained by choosing for m a value greater than the critical integer ; but it is usually most advantageous to take m equal to its least lawful value. Ex. 1. The singularities of the function ?r cot 772 are given by z = \, for all integer values of X from — oo to +00 including zero, so that the origin is a singularity. The integral to be considered is - 1 M IT cot vt fz\m ,, = ~ — . I — - (- ) at. 2iri J t-z \tj We take the contour to be a circle of very large radius R chosen so that the circumference does not pass innuitesimally near any one of the singularities of TT coint at infinity; this 79.] AN INFINITE SERIES OF FUNCTIONS 125 is, of course, possible because there is a finite distance between any two of them. Then, round the circumference so taken, n cot nt is never infinite : hence its modulus is never greater than some finite quantity M. Let t = Reei, so that ~=id6; then v and therefore Z .--—. t-z for some point t on the circle. Now, as the circle is very large, we have \t-z\ infinite : hence \J\ can be made zero merely by taking m unity. Thus, for the function TT cot TTZ, the critical integer is unity. Hence from the general theorem we have the equation 1 fir cot nt z j 7T COt 772= -5— . 2 I— -dt, 2TTI J t-Z t the summation extending to all the points X for integer values of X = - oc to + oo , and each integral being taken round a small circle centre X. -vr . » . 1 /"(*) TT cot irt z , Now if, in -— . • -dt. 2m J t - z t we take t=\ + (, we have where P(Q = 0 when £= 0; and therefore the value of the integral is •*./ (*-*+{) (x+fl t In the limit when |f| is infinitesimal, this integral z = (X-2)X 1 1 ~X-2 X' and therefore /*. (z) = -J— + 1 A ' z-X X' if X be not zero. And for the zero of X, the value of the integral is (p 126 REGION OF CONTINUITY [79. so that F0(z) is -. In fact, in the notation of § 72, we have z o P-A»JL ^ \z-\J~z-\' arid the expansion of GK needs to be carried only to one term. 1 A=ao /I 1\ We thus have 7rcot7rs = — f- 2 — N+=r)> z A=-co \Z-X A/ the summation not including the zero value of X. Ex. 2. Obtain, ab initio, the relation SHI2 3 A=_aj (z-X7r)2' p. 3. Shew that, if 1 °° 1 1 then "-^^ = - + 2z 2 ^3-^1- R(z) z i=lR(\)z*-\* (Gylden, Mittag-Leffler.) Ex. 4. Obtain an expression, in the form of a sum, for IT cot irz where Q(z) denotes (1 -z) (l -^ (l -|J ...... ^-j)*- 80. The results obtained in the present chapter relating to functions which have an unlimited number of singularities, whether distributed over the whole plane or distributed over only a finite portion of it, shew that analytical functions can be represented, not merely as infinite converging series of powers of the variable, but also as infinite converging series of functions of the variable. The properties of functions when represented by series of powers of the variable depended in their proof on the condition that the series proceeded in powers; and it is therefore necessary at least to revise those properties in the case of functions when represented as series of functions of the variable. Let there be a series of uniform functions /i (z), /, (z), . . . ; then the aggregate of values of z, for which the series 1*1 has a finite value, is the region of continuity of the series. If a positive quantity p can be determined such that, for all points z within the circle z — a\ = p, 80.] OF A SERIES OF FUNCTIONS 127 00 the series 2 fi(z) converges uniformly and unconditionally*, the series is said to converge in the vicinity of a. If R be the greatest value of p for which this holds, then the area within the circle z — a\ = R is called the domain of a; and the series converges uniformly and uncon ditionally in the vicinity of any point in the domain of a. It will be proved in § 82 that the function can be represented by power- series, each such series being equivalent to the function within the domain of some one point. In order to be able to obtain all the power-series, it is necessary to distribute the region of continuity of the function into domains of points where it has a uniform, finite value. We therefore form the domain of a point 6 in the domain of a from a knowledge of the singularities of the function, then the domain of a point c in the domain of 6, and so on ; the aggregate of these domains is a continuous part of the plane which has isolated points and which has one or several lines for its boundaries. Let this part be denoted by At. For most of the functions, which have already been considered, the region A1} thus obtained, is the complete region of continuity. But examples will be adduced almost immediately to shew that A-^ does not necessarily include all the region of continuity of the series under consideration. Let a' be a point not in A-^ within whose vicinity the function has a uniform, finite value ; then a second portion A2 can be separated from the whole plane, by proceeding from a' as before from a. The limits of A± and A2 may be wholly or partially the same, or may be independent of one another : but no point within either can belong to the other. If there be points in the region of con tinuity which belong to neither A1 nor A2, then there must be at least another part of the plane A3 with properties similar to At and^l2- And so on. The 00 series 2 fi(z) converges uniformly and unconditionally in the vicinity of »=i « every point in each of the separate portions of its region of continuity. It was proved that a function represented by a series of powers has a definite finite derivative at every point lying actually within the circle of convergence of the series, but that this result cannot be affirmed for a point on the boundary of the circle of convergence even though the value of the series itself should be finite at the point, an illustration being provided by the hypergeometric series at a point on the circumference of its circle of * In connection with most of the investigations in the remainder of this chapter, Weierstrass's memoir " Zur Functionenlehre " already quoted (p. 112, note) should be consulted. It may be convenient to give here Weierstrass's definition (I.e., p. 70) of uniform, unconditional convergence. A series 2 fn converges uniformly, if an integer m can be determined so that /» can be made less than any arbitrary positive quantity, however small ; and it converges uncon ditionally, if the uniform convergence of the series be independent of any special arrangement of order or combination of the terms. 128 REGION OF CONTINUITY OF [80. convergence. It will appear that a function represented by a series of functions has a definite finite derivative at every point lying actually within its region of continuity, but that the result cannot be affirmed for a point on the boundary; and an example will be given (§ 83) in which the derivative is indefinite. Again, it has been seen that a function, initially defined by a given power- series, is, in most cases, represented by different analytical expressions in different parts of the plane, each of the elements being a valid expression of the function within a certain region. The questions arise whether a given analytical expression, either a series of powers or a series of functions : (i) can represent different functions in the same continuous part of its region of continuity, (ii) can represent different functions in distinct (that is, non- continuous) parts of its region of continuity. 81. Consider first a function defined by a given series of powers. Let there be a region A' in the plane and let the region of continuity of the function, say g (z), have parts common with A'. Then if a0 be any point in one of these common parts, we can express g {z) in the form P (z — a0) in the domain of a0. As already explained, the function can be continued from the domain of a0 by a series of elements, so that the whole region of continuity is gradually covered by domains of successive points ; to find the value in the domain of any point a, it is sufficient to know any one element, say, the element in the domain of a0. The function is the same through its region of continuity. Two distinct cases may occur in the continuations. First, it may happen that the region of continuity of the function g (z) extends beyond A'. Then we can obtain elements for points outside A', their aggregate being a uniform analytical function. The aggregate of elements then represents within A' a single analytical function : but as that function has elements for points without A, the aggregate within A' does not completely represent the function. Hence If a function be defined within a continuous region of a plane by an aggregate of elements in the form of power-series, which are continuations of one another, the aggregate represents in that part of the plane one (and only one) analytical function : but if the power-series can be continued beyond the boundary of the region, the aggregate of elements within the region is not the complete representation of the analytical function. This is the more common case, so that examples need not be given. Secondly, it may happen that the region of continuity of the function does not extend beyond A' in any direction. There are then no elements of the function for points outside A' and the function cannot be continued beyond the boundary of A. The aggregate of elements is then the complete representation of the function and therefore : 81.] A SERIES OF POWERS 129 If a function be defined within a continuous region of a plane by an aggregate of elements in the form of power-series, which are continuations of one another, and if the power-series cannot be continued across the boundary of that region, the aggregate of elements in the region is the complete representa tion of a single uniform monogenic function which exists only for values of the variable within the region. The boundary of the region of continuity of the function is, in the latter case, called the natural limit of the function*, as it is a line beyond which the function cannot be continued. Such a line arises for the series l + 2z + ^ + 2z9 + ... , in the circle \z = 1, a remark due to Kronecker; other illustrations occur in connection with the modular functions, the axis of real variables being the natural limit, and in connection with the automorphic functions (see Chapter XXII.) when the fundamental circle is the natural limit. A few examples will be given at the end of the present Chapter. It appears that Weierstrass was the first to announce the existence of natural limits for analytic functions, Berlin Monatsber. (1866), p. 617 ; see also Schwarz, Ges. Werke, t. ii, pp. 240 — 242, who adduces other illustrations and gives some references ; Klein and Fricke, Vorl. uber die Theorie der elliptischen Modulfunctioncn, t. i, (1890), p. 110; Jordan, Cows d' Analyse, t. iii, pp. 609, 610. Some interesting examples and discussions of functions, which have the axis of real variables for a natural limit, are given by Hankel, " Untersuchungen liber die unendlich oft oscillirenden und unstetigen Functionen," Math. Ann., t. xx, (1870), pp. 63—112. 82. Consider next a series of functions of the variable ; let it be The region of continuity may be supposed to consist of several distinct parts, in the most general case ; let one of them be denoted by A. Take some point in A, say the origin, which is either an ordinary point or an isolated singularity; and let two concentric circles of radii R and R' be drawn in A, so that R < z =r<R, and the space between these circles lies within A. In this space, each term of the series is finite and the whole series converges uniformly and uncon ditionally. Now let fi (z) be expanded in a series of powers of z, which series con verges within the space assigned, and in that expansion let ^ be the co- oo efficient of z* ; then we can prove that 2 i^ is finite and that the series ( / °° \ s |(sO n. (\i = 0 I * Die natiirliche Grenze, according to German mathematicians. F. 130 REGION OF CONTINUITY [82. converges uniformly and unconditionally within this space, so that •x. (/ oo 2 /,(*) = 2 2 i=l " /A {\i=Q 00 Because the infinite series 2 fi (z) converges uniformly and uncon ditionally, a number n can be chosen so that where & is an arbitrary finite quantity, ultimately made infinitesimal; and therefore also i=n where n' > n and is infinite in the limit. Now since the number of terms in the series is not infinite before the limit, we have But the original series converges unconditionally, and therefore k is not less n than the greatest value of the modulus of 2 fi(z) for points within the i=n region; hence, by § 29, we have n 2 V < AT <i. »•=» 00 Moreover, A; is not less than the greatest value of the modulus of 2 fi(z) in the given region ; and so 00 2 i^ < AT *. i=n Now, by definition, k can be made as small as we desire by choice of n ; hence the series is a converging series. Let it be denoted by A^. n-l oo Let 2 r'M = A /, 2 ifj, = A M" ; then, by the above suppositions, we can always choose n so that k being any assignable small quantity. 82.] OF A SERIES OF FUNCTIONS 131 When two new quantities r± and r2 are introduced, as in § 28, satisfying the inequalities f-f ^ ly ^ \ iv --• /y» ^ 7?' -il/<^/l<s.|.S|<i./2<.-fl, the integer w can be chosen so that \Ap'\ < kr~* < kr^. f- r. Then and so that 2 .— 00 2 - - <k M=-oo r — r-i r2-r Hence the series 2 A^'z^ can by choice of n be made to have a modulus less than any finite quantity ; and therefore, since /u.= oo n — 1 (for there is a finite number of terms in the coefficients on each side, the expansions are converging series, and the sum on the right-hand side is a finite quantity), it follows that the series converges uniformly. Finally, we have 2 . fl= —00 2 ft (*) - 24^ = 2 /< (z) - <=i 1=1 and therefore 2 t'=n r ~ which, as k can be diminished indefinitely, can be made less than any finite jlX=00 quantity. Hence the series 2 A^ converges unconditionally, and there- fi= —00 fore we have provided 00 jlt=00 2 /;(*)= 2 . l'=l /u= — oo 9—2 132 REGION OF CONTINUITY [82. When we take into account all the parts of the region of continuity of the series, constituted by the sum of the functions, we have similar expansions in the form of successive series of powers of z — c, converging uniformly and unconditionally in the vicinities of the successive points c. But, in forming the domains of these points c, the boundary of the region of continuity of the function must not be crossed ; and a new series of powers is required when the circle of convergence of any one series (lying within the region of continuity) is crossed. It therefore appears that a converging series of functions of a variable can be expressed in the form of series of powers of the variable which converge within the parts of the plane where the series of functions converges uniformly and unconditionally ; but the equivalence of the two expressions is limited to such parts of the plane and cannot be extended beyond the boundary of the region of continuity of the series of functions. If the region of continuity of a series of functions consist of several parts of the plane, then the series of functions can in each part be expressed in the form of a set of converging series of powers : but the sets of series of powers are not necessarily the same for the different parts, and they are not necessarily continuations of one another, regarded as power-series. Suppose, then, that the region of continuity of a series of functions F(z)=lfi(z) i=l consists of several parts A1} A.2, Within the part A^ let F (z) be represented, as above, by a set of power-series. At every point within A1} the values of F(z) and of its derivatives are each definite and unique ; so that, at every point which lies in the regions of convergence of two of the power-series, the values which the two power-series, as the equivalents of F (z) in their respective regions, furnish for F (z) and for its derivatives must be the same. Hence the various power-series, which are the equivalents of F (z) in the region Aly are continuations of one another: and they are sufficient to determine a uniform monogenic analytic function, say F^ (z}. The functions F(z) and Fl(z) are equivalent in the region Al; and therefore, by § 81, the series of functions represents one and the same function for all points within one continuous part of its region of continuity. It may (and frequently does) happen that the region of continuity of the analytical function F± (z) extends beyond A± ; and then F-^ (z) can be continued beyond the boundary of A^ by a succession of elements. Or it may happen that the region of continuity of Fl (z) is completely bounded by the boundary of A^ ; and then the function cannot be continued across that boundary. In either case, the equivalence 00 of F-L(Z) and 2 fi(z) does not extend beyond the boundary of Alt one 82.] OF A SERIES OF FUNCTIONS 133 00 complete and distinct part of the region of continuity of 2 fi(z); and i = \ therefore, by using the theorem proved in § 81, it follows that : A series of functions of a variable, which converges within a continuous part of the plane of the variable z, is either a partial or a complete representation of a single uniform, analytic function of the variable in that part of the plane. 83. Further, it has just been proved that the converging series of functions can, in any of the regions A, be changed into an equivalent uniform, analytic function, the equivalence being valid for all points in that region, say 2 /(•). 4(4 i = l But for any point within A, the function Fl(z) has a uniform finite derivative oo (§ 21); and therefore also 2 fi(z) has a uniform finite derivative. The i=l equivalence of the analytic function and the series of functions has not been proved for points on the boundary; even if they are equivalent there, the function I\ (z) cannot be proved to have a uniform finite derivative at every 00 point on the boundary of A, and therefore it cannot be affirmed that 2 ft (z) i=\ has, of necessity, a uniform, finite derivative at points on the boundary of A, even oo though the value of 2 fi(z) be uniform and finite at every point on the i=l boundary*. Ex. In illustration of the inference just obtained, regarding the derivative of a function at a point on the boundary of its region of continuity, consider the series g(z)= 2 &V", n=0 where b is a positive quantity less than unity, and a is a positive quantity which will be taken to be an odd integer. For points within and on the circumference of the circle \z =1, the series converges uniformly and unconditionally; and for all points without the circle the series diverges. It thus defines a function for points within the circle and on the circumference, but not for points without the circle. Moreover for points actually within the circle the function has a first derivative and consequently has any number of derivatives. But it cannot be declared to have a derivative for points on the circle: and it will in fact now be proved that, if a certain condition be satisfied, the derivative for variations at any point on the circle is not merely infinite but that the sign of the infinite value depends upon the direction of the variation, so that the function is not monogenic for the circumference t. * It should be remarked here, as at the end of § 21, that the result in itself does not contravene Biemann's definition of a function, according to which (§ 8) -^ must have the same value what ever be tbe direction of the vanishing quantity dz ; at a point on the boundary of the region there are outward directions for which die is not defined. t The following investigation is due to Weierstrass, who communicated it to Du Bois-Eeymond : see Crclle, t. Ixxix, (1875), pp. 29—31. 134 A SERIES OF FUNCTIONS [83. Let z = eei: then, as the function converges unconditionally for all points along the circle, we take f(ff)= 2 lnea"ei, 71=0 where 6 is a real variable. Hence m-l IV,an(0 + 4>)*_,,«WWl = s«nH - — - H=O 1 an$ J /•ea">+»> (0 + <f>) i _ ea™+«0(S + 2 &w + M - -T - 1 1 «=o I 9 J assuming m, in the first place, to be any positive integer. To transform the first sum on the right-hand side, we take and therefore pan (0 + <j>) i _ a"0i 2 (ab}n n=0 <M21^n 8Jn(fr-*) if ab>\. Hence, on this hypothesis, we have 2 (ab)n \ — \ =y r i » *=o ( a"0 J ao - 1 where 7 is a complex quantity with modulus <1. To transform the second sum on the right-hand side, let the integer nearest to am be am, so that 7T for any value of m : then taking we have \tr^-x> — %n, and cos x is not negative. We choose the quantity <f> so that and therefore TT am ff) — — — , 0 which, by taking m sufficiently large (a is > 1), can be made as small as we please. We now have am+"(6 +<i>)i = Qaniti (1 + o™) _ _ / _ j N°™ if a be an odd integer, and _ am+nOi _ ani (x + iram] _ / _ j \<»meana;i , a"xi Hence CD /• and therefore 2 &- + « f ,,=0 i _ - ( - 1) 2 6" 83.] MAY NOT POSSESS A DERIVATIVE 135 The real part of the series on the right-hand side is 2 bn{l + cosanx}; n=0 every term of this is positive and therefore, as the first term is 1 + cos x, the real part > 1+cos.r >1 for cos x is not negative ; and it is finite, for it is <2 2 bn K=0 2 <r^6- Moreover far < TT — x < frr, so that -- is positive and >-. Hence TT — x 6 where TJ is a finite complex quantity, the real part of which is positive and greater than unity. We thus have where |y'|<l, and the real part of 77 is positive and > 1. Proceeding in the same way and taking IT ' am ' TT+X so that % = — — , we find — — — — t_LJ — _ ( _ iy™ (a^ where |y/|<l and the real part of TJ^ a finite complex quantity, is positive and greater than unity. If now we take ab - 1 > fn-, the real parts of - — + y -*-—= , say of f, O 7T (tO — 1 and of |li+yi'__L_,sayof fl, are both positive and different from zero. Then, since and ~x- = (_!)«- (ab)m d , /(. m being at present any positive integer, we have the right-hand sides essentially different quantities, because the real part of the first is of sign opposite to the real part of the second. Now let m be indefinitely increased; then $ and x are infinitesimal quantities which ultimately vanish ; and the limit of - [/(# + </>)-/(#)] for $ = 0 is a complex infinite 136 ANALYTICAL EXPRESSION [83. quantity with its real part opposite in sign to the real part of the complex infinite quantity which is the limit of $ — ~^ f°r = ®- If# had a differential coefficient A these two limits would be equal : hence / (0) has not, for any value of 6, a determinate differential coefficient. From this result, a remarkable result relating to real functions may be at once derived. The real part of / (<9) is 2 6ncos(an<9), n=0 which is a series converging uniformly and unconditionally. The real parts of -(-ir («&)-<: and of +(-l)am(a6)TOf1 are the corresponding magnitudes for the series of real quantities : and they are of opposite signs. Hence for no value of 6 has the series 2 6"cos(an<9) n=0 a determinate differential coefficient, that is, we can choose an increase <£ and a decrease ^ of 6, both being made as small as we please and ultimately zero, such that the limits of the expressions 0 -X are different from one another, provided a be an odd integer and ab > 1 +|TT. The chief interest of the above investigation lies in its application to functions of real variables, continuity in the value of which is thus shewn not necessarily to imply the existence of a determinate differential coefficient defined in the ordinary way. The application is due to Weierstrass, as has already been stated. Further discussions will be found in a paper by Wiener, Crelle, t. xc, (1881), pp. 221 — 252, in a remark by Weierstrass, Abh. aus der Functionenlehre, (1886), p. 100, and in a paper by Lerch, Crelle, t. ciii, (1888), pp. 126 — 138, who constructs other examples of continuous functions of real variables ; and an example of a continuous function without a derivative is given by Schwarz, Ges. Werke, t. ii, pp. 269 — 274. The simplest classes of ordinary functions are characterised by the properties : — (i) Within some region of the plane of the variable they are uniform, finite and continuous : (ii) At all points within that region (but not necessarily on its boundary) they have a differential coefficient : (iii) When the variable is real, the number of maximum values and the number of minimum values within any given range is finite. The function 2 bn cos (anQ\ suggested by Weierstrass, possesses the first but not the 71=0 second of these properties. Kb'pcke (Math. Ann., t. xxix, pp. 123 — 140) gives an example of a function which possesses the first and the second but not the third of these properties. 84. In each of the distinct portions Alt A.2>... of the complete region of continuity of a series of functions, the series can be represented by a monogenic analytic function, the elements of which are converging power- series. But the equivalence of the function -series and the monogenic 84.] REPRESENTING DIFFERENT FUNCTIONS 137 analytic function for any portion A^ is limited to that region. When the monogenic analytic function can be continued from A^ into Az, the continua tion is not necessarily the same as the monogenic analytic function which is 00 the equivalent of the series 2 fi(z) in A2. Hence, if the monogenic analytic i = l functions for the two portions A^ and A2 be different, the function-series represents different functions in the distinct parts of its region of continuity. A simple example will be an effective indication of the actual existence of such variety of representation in particular cases ; that, which follows, is due to Tannery*. Let a, b, c be any three constants ; then the fraction a + bczm Y+'bzm ' when m is infinite, is equal to a if z \ < 1, and is equal to c if | z > 1. Let m0, m1} m2>... be any set of positive integers arranged in ascending order and be such that the limit of mn, when n = oo , is infinite. Then, since a + bczm* a + bczm° » {a + bczmi a + bczm 1 + bzm» 1 -f bzm° f.i (1 + bzmi I + bz'" ^mo " ~* a) the function <f)(z), defined by the equation ,. a + bczm° ., N S f 0^-^-1-1)^-1 + (z} = TT6^ + b (G ~ a) £ {(I + bz^) (i + 6^ converges uniformly and unconditionally to a value a if \ z < 1, awe? converges uniformly and unconditionally to a value c if z \ > 1. But it does not con verge uniformly and unconditionally if z \ = 1. The simplest case occurs when b = — 1 and m^ = 2* ; then, denoting the function by <f> (z), we have a - cz , . ( z z2 z4 that is, the function <f> (z) is equal to a if z < 1, and it is equal to c if * It is contained in a letter of Tannery's to Weierstrass, who communicated it to the Berlin Academy in 1881, Abh. aus der Functionenlehre, pp. 103, 104. A similar series, which indeed is equivalent to the special form of $ (z), was given by Schroder, Schlfim. Zeitschrift, t. xxii, (1876), p. 184; and Pringsheim, Math. Ann., t. xxii, (1883), p. 110, remarks that it can be deduced, without material modifications, from an expression given by Seidel, Crelle, t. Ixxiii, (1871), pp. 297- -299. 138 LINE OF SINGULARITIES [84. When \z =\, the function can have any value whatever. Hence a circle of radius unity is a line of singularities, that is, it is a line of discontinuity for the series. The circle evidently has the property of dividing the plane into two parts such that the analytical expression represents different functions in the two parts. If we introduce a new variable £ connected with z by the relation* l +z then, if £= £ + iy and z = x + iy, we have 1 rfS. nil fc i — x y so that £ is positive when \z\< 1, and £ is negative when \ z \ > 1. If then the function %(£) is equal to a or to c according as the real part of f is positive or negative. And, generally, if we take £ a rational function of z and denote the modified form of </> (£), which will be a sum of rational functions of z, by ^(z), then <f>i(z) will be equal to a in some parts of the plane and to c in other parts of the plane. The boundaries between these parts are lines of singular points : and they are constituted by the ^-curves which correspond to £| = 1. 85. Now let F(z) and G(z) be two functions of z with any number of singularities in the plane : it is possible to construct a function which shall be equal to F (z} within a circle centre the origin and to G (z) without the circle, the circumference being a line of singularities. For, when we make a = 1 and c = 0 in </> (z) of § 84, the function 1 z z* z4 00)=- -- + -. — r + •-: — : + -. -^r + . . . V/ 1—0 Z2 — I Z*— I ZS —I is unity for all points within the circle and is zero for all points without it : and therefore G(z} + {F(z)-G(z)}6(z} is a function which has the required property. Similarly F3 (z) + {F, (z) - F, (z)} 6 (z) + {F, (z) - F3 (z}} 6 ( is a function which has the value Fl (z) within a circle of radius unity, the value F2 (z) between a circle of radius unity and a concentric circle of radius r greater than unity, and the value F3(z) without the latter circle. All the singularities of the functions F1} F2, F3 are singularities of the function thus represented; and it has, in addition to these, the two lines of singularities given by the circles. * The significance of a relation of this form will be discussed in Chapter XIX. 85.] MONOGENIC FUNCTIONALITY 139 Again, 6 is a function of s, which is equal to F(z) on the positive side of the axis of y, and is equal to G (z) on the negative side of that axis. 1+2 Also, if we take £e l —p\ = ^~i where ax and p1 are real constants, as an equation defining a new variable £ + iy, we have | cos at + 77 sin aj -pl = p. \23T~2 so that the two regions of the 2-plane determined by \z\<l and \z\>l correspond to the two regions of the {"-plane into which the line £ cos a: + 77 sin al—p1 = 0 divides it. Let ,-«'ai — », — 1\ so that on the positive side of the line £ cos at + 77 sin aj — p1 = 0 the function 6l is unity and on the negative side of that line it is zero. Take any three lines defined by ax, p1; a2, p2', a,, pn respectively ; then AJ.A11 (2)\-F/(l) is a function which has the value F within the triangle, the value - F in three of the spaces without it, and the value zero in the remaining three spaces without it, as indi cated in the figure (fig. 13). And for every division of the plane by lines, into which a circle can be transformed (3) by rational equations, as will be explained when conformal representation is discussed (1) / hereafter, there is a possibility of represent- Fig. 13. ing discontinuous functions, by expressions similar to those just given. These examples are sufficient to lead to the following result*, which is complementary to the theorem of § 82 : When the region of continuity of an infinite series of functions consists of several distinct parts, the series represents a single function in each part but it does not necessarily represent the same function in different parts. It thus appears that an analytical expression of given form, which con verges uniformly and unconditionally in different parts of the plane separated from one another, can represent different functions of the variable in those different parts ; and hence the idea of monogenic functionality of a complex variable is not coextensive with the idea of functional dependence expressible through arithmetical operations, a distinction first established by Weierstrass. 86. We have seen that an analytic function has not a definite value at an essential singularity and that, therefore, every essential singularity is excluded from the region of definition of the function. * Weierstrass, I.e., p. 90. 140 SINGULAR LINES [86. Again, it has appeared that not merely must single points be on occasion excluded from the region of definition but also that functions exist with continuous lines of essential singularities which must therefore be excluded. One method for the construction of such functions has just been indicated : but it is possible to obtain other analytical expressions for functions which possess what may be called a singular line. Thus let a function have a circle of radius c as a line of essential singularity*; let it have no other singularities in the plane and let its zeros be al} a2, a3,..., supposed arranged in such order that, if pneie" = an> then I Pn C | ^ Pn+i ~ C > so that the limit of pn, when n is infinite, is c. Let cn = ceie«, a point on the singular circle, corresponding to an which is assumed not to lie on it. Then, proceeding as in Weierstrass's theory in § 51, if «.= oo („ _ Gz= n where gn(z) = - + L_ +... + _ Z-Cn 2 \Z-CnJ mn - I \ Z - Cn G (z) is a uniform function, continuous everywhere in the plane except along the circumference of the circle which may be a line of essential singularities. Special simpler forms can be derived according to the character of the series of quantities constituted by | an - cn . If there be a finite integer m, 00 such that 2 an — cn m is a converging series, then in gn (z) only the first M = l m — 1 terms need be retained. Ex. Construct the function when m being a given positive integer and r a positive quantity. Again, the point cn was associated with an so that they have the same argument : but this distribution of points on the circle is not necessary and can be made in any manner which satisfies the condition that in the limited 00 case just quoted the series 2 an — cn m is a converging series. Singular lines of other classes, for example, sectioiis\ in connection with functions defined by integrals, arise in connection with analytical functions. They are discussed by Painleve, "Sur les lignes singulieres des fonctions analytiques," (These, Gauthier- Villars, Paris, 1887). Ex. Shew that, if the zeros of a function be the points . _b+c— (a — d) i ZT ^ ~7 i 7T \ • 5 * This investigation is due to Picard, Comptes Rendus, t. xci, (1881), pp. 690—692. t Called conpures by Hermite ; see § 103. 86.] LACUNARY FUNCTIONS 141 where a, ?;, c, d are integers satisfying the condition ad-bo = l, so that the function has a circle of radius unity for an essential singular line, then if b + di „ 2J = -^ - =— , , d+bi' ( A the function n \ 5 e z (z — li where the product extends to all positive integers subject to the foregoing condition ad-bc = l, is a uniform function finite for all points in the plane not lying on the circle of radius unity. (Picard.) 87. In the earlier examples, instances were given of functions which have only isolated points for their essential singularities : and, in the later examples, instances have been given of functions which have lines of essential singularities, that is, there are continuous lines for which the functions do not exist. We now proceed to shew how functions can be constructed which do not exist in assigned continuous spaces in the plane, these spaces being aggregates of essential singularities. Weierstrass was the first to draw attention to lacunary functions, as they may be called ; the following investigation in illustration of Weierstrass's theorem is due to Poincare' *. Take any convex curve in the plane, say G ; and consider the function *z^b' where the quantities A are constants, subject to the conditions (i) The series ^\A\ converges uniformly and unconditionally : (ii) Each of the points b is either within or on the curve G : (iii) The points b are the aggregate of all rational j points within and on C : then the function is a uniform analytical function for all points without C and it has the area of G for a lacunary space. First, it is evident that, if z = b, then the series does not converge. Moreover as the points b are the aggregate of all the rational points within or on C, there will be an infinite number of singularities in the immediate vicinity of b : we shall thus have an unlimited number of terms each infinite of the first order, and thus (§ 42) the point b will be an essential singularity. As this is true of all points z within or on C, it follows that the area C is a lacunary space for the function, if the function exist at all. Secondly, let z be a point without G ; and let d be the distance of z from the nearest point of the boundary of C^f% so that d is not a vanishing quantity. * Acta Soc. Fenn., t. xii, (1883), pp. 341—350. J Rational points within or on C are points whose positions can be determined rationally in terms of the coordinates of assigned points on C ; examples will be given. t This will be either the shortest normal from z to the boundary or the distance of z from some point of abrupt change of direction, as for instance at the angular point of a polygon. 142 FUNCTIONS WITH [87. Then | z — b \ ^ d ; and therefore A _ \A\ \A\ ~\z-b\< d ' z-b so that -b A z-b Now 2 j.A| converges uniformly and unconditionally and therefore, as d does not vanish, z-b converges uniformly and unconditionally, that is. is a function of 2 which converges uniformly and unconditionally for every point without C. Let it be denoted by <£ (z). Let c be any point without C, and let r be the radius of the greatest circle centre c which can be drawn so as to have no point of C within itself or on its circumference, so that r is the radius of the domain of c; then b — c > r, for all points b. If we take a point z within this circle, we have \z — c =6r, where 6 < 1. Now for all points within this circle the function <£ {z} converges uniformly, A and every term -- =• of <f> (z) is finite. Also, for points within the circle, we A can expand -- j in powers of z — c in the form of a converging series. Hence, by § 82, we have <£(*)= 2 Bm(z-c)m, a series converging uniformly and unconditionally for all points within the circle centre c and radius r, which circle is the circle of convergence of the series. The function can be expressed in the usual manner over the whole of the region of continuity, which is the part of the plane without the curve C. Thus 0 (z) is a uniform analytical function, having the area of C for a lacunary space. As an example, take a convex polygon having o1} ...... , ap for its angular points; then any point ...... +mj>ap TOI + ...... +mp where mlt ...... , mp are positive integers or zero (simultaneous zeros being excluded), is 87.] LACUNARY SPACES 143 either within the polygon or on its boundary : and any rational point within the polygon or on its boundary can be represented by p 2 mrar r=l P ' 2 mr r=l by proper choice of ?n15 ...... , mp, a choice which can be made in an infinite number of ways. Let ult ...... , Up be given quantities, the modulus of each of which is less than unity: then the series •9-11 m> 11 mf «& ^ ' I ...... ftp o converges uniformly and unconditionally. Then all the assigned conditions are satisfied for the function _ .. . + mpap > ' ml + ...... +mp J and therefore it is a function which converges uniformly and unconditionally everywhere outside the polygon and which has the polygonal space (including the boundary) for a lacunary space. If, in particular, p = Z, we obtain a function which has the straight line joining ax and a2 as a line of essential singularity. When we take at = 0, a.2 = 1 and slightly modify the summation, we obtain the function 2 2 ^ 2 w=l m=0 W& 7i which, when u^ <\ and |w2|<l, converges uniformly and unconditionally everywhere in the plane except at points between 0 and 1 on the axis of real quantities, this part of the axis being a line of essential singularity. For the general case, the following remarks may be made : (i) The quantities u1} u2>... need not be the same for every term; a numerator, quite different in form, might be chosen, such as (mj2+ ... + m/)"'1 where 2//, > p ; all that is requisite is that the series, made up of the numerators, should converge uniformly and unconditionally. (ii) The preceding is only a particular illustration and is not necessarily the most general form of function having the assigned lacunary space. It is evident that the first step in the construction of a function, which shall have any assigned lacunary space, is the formation of some expression which, by the variation of the constants it contains, can be made to represent indefinitely nearly any point within or on the contour of the space. Thus for the space between two concentric circles of radii a and c and centre the origin we should take Wja + O-WjU ^a« -a£- e n n 144 EXAMPLES [87. which, by giving m^ all values from 0 to n, ra2 all values from 0 to n — 1 and n all values from 1 to infinity will represent all rational points in the space : and a function, having the space between the circles as lacunary, would be given by oo n n-1 2 2 2 n=l »»!=(> m2=0 (n — raj) b ^ 271- •r /3 .6 — C n provided u\ < 1, u^ < 1, u2 < 1. In particular, if a = 6, then the common circumference is a line of essential singularity for the corresponding function. It is easy to see that the function z — ae n ao 2n-l m n provided the series 2 2 u v n=l m=0 m,n m, n converges uniformly and unconditionally, is a function having the circle |0| = a as a line of essential singularity. Other examples will be found in memoirs by Goursat*, Poincaref, and HomenJ. Ex. 1. Shew that the function where r is a real positive quantity and the summation is for all integers m and n between the positive and the negative infinities, is a uniform function in all parts of the plane except the axis of real quantities which is a line of essential singularity. Ex. 2. Discuss the region in which the function w=i m=i jf/=i i ^- . ••- • 2—1 1 i \7i 71 is definite. (Homen.) Ex. 3. Prove that the function n=0 exists only within a circle of radius unity and centre the origin. (Poincare.) Ex. 4. An infinite number of points at, a2, as, are taken on the circumference of a given circle, centre the origin, so that they form the aggregate of rational points on the circumference. Shew that the series 2 l Z can be expanded in a series of ascending powers of z which converges for points within the circle, but that the function cannot be continued across the circumference of the circle. (Stieltjes.) * Comptes Rendus, t. xciv, (1882), pp. 715—718 ; Bulletin de Darboux, 2me Ser. , t. xi, (1887), pp. 109—114. t In the memoir, quoted p. 138, and Comptes Rendus, t. xcvi, (1883), pp. 1134-1136. + Acta Soc. Fenn., t. xii, (1883), pp. 445—464. 87.] EXAMPLES 145 Ex. 5. Prove that the series 2 | : 7T .00 -" K1-2TO- 9 oo 22 ~li)2) ' 7T _oo _oo (^(1 — 2wi — 2nz i) \zm-\-Anz~ where the summation extends over all positive and negative integral values of ra and of n except simultaneous zeros, is a function which converges uniformly and unconditionally for all points in the finite part of plane which do not lie on the axis of y ; and that it has the value +1 or - 1 according as the real part of z is positive or negative. (Weierstrass.) Ex. 6. Prove that the region of continuity of the series consists of two parts, separated by the circle z\ = l which is a line of infinities for the series : and that, in these two parts of the plane, it represents two different functions. _<a'ir If two complex quantities a> and to' be taken, such that z = e ^ and the real part of ^. is positive, and if they be associated with the elliptic function $ (u) as its half-periods, then for values of z which lie within the circle z = \ in the usual notation of Weierstrass's theory of elliptic functions. Find the function which the series represents for values of z without the circle \z\ = \. (Weierstrass.) Ex. 7. Four circles are drawn each of radius -^ having their centres at the points 1, i, - 1, -i respectively; the two parts of the plane, excluded by the four circumferences, are denoted the interior and the exterior parts. Shew that the function n='K sini^TT ( 1 1 1 1 is equal to IT in the interior part and is zero in the exterior part. (Appell.) Ex. 8. Obtain the values of the function »;-l- (-!)•(, i >1 l «=i n V1 •> (2 + l)« (2-l)«J in the two parts of the area within a circle centre the origin and radius 2 which lie without two circles of radius unity, having their centres at the points 1 and - 1 respectively. (Appell.) Ex- 9- If and ,~3 ...... amr (2-«m)3 J where the regions of continuity of the functions F extend over the whole plane, then / (z) is a function existing everywhere except within the circles of radius unity described round the points a, , «2, ...... , an. (Teixeira.) F- 10 146 CLASSIFICATION [87. Ex. 10. Let there be n circles having the origin for a common centre, and let £,, (72, ...... , (7n, C'n + 1 be % + 1 arbitrary constants; also let a1} a2, ...... , an be any w points lying respectively on the circumferences of the first, the second, ...... , the nth circles. Shew that the expression 1 ("(CL 27T./0 W* has the value <7m for points z lying between the (»w - l)th and the with circles and the value (7n + 1 for points lying without the nth circle. Construct a function which shall have any assigned values in the various bands into which the plane is divided by the circles. (Pincherle.) 88. In § 32 it was remarked that the discrimination of the various species of essential singularities could be effected by means of the properties of the function in the immediate vicinity of the point. Now it was proved, in § 63, that in the vicinity of an isolated essential singularity b the function could be represented by an expression of the form for all points in the space without a circle centre b of small radius and within a concentric circle of radius not large enough to include singularities at a finite distance from b. Because the essential singularity at b is isolated, the radius of the inner circle can be diminished to be all but infinitesimal : the series P (z — b) is then unimportant compared with G I —31 ) , which can be regarded as characteristic for the singularity of the function. Another method of obtaining a function, which is characteristic of the singularity, is provided by § 68. It was there proved that, in the vicinity of an essential singularity a, the function could be represented by an expression of the form where, within a circle of centre a and radius not sufficiently large to include the nearest singularity at a finite distance from a, the function Q (z — a) is finite and has no zeros : all the zeros of the given function within this circle (except such as are absorbed into the essential singularity at a) are zeros of the factor H ( - - ] , and the integer-index n is affected by the number of these zeros. When the circle is made small, the function z-a can be regarded as characteristic of the immediate vicinity of a or, more briefly, as characteristic of a. 88.] OF SINGULARITIES 147 It is easily seen that the two characteristic functions are distinct. For if F and F^ be two functions, which have essential singularities at a of the same kind as determined by the first characteristic, then F(z)-Fl(z) = P(z-a)-Pl(z-a) = P,(z-a\ while if their singularities at a be of the same kind as determined by the second characteristic, then F(z)_Q(z-a) f\(*)-Q^-~a) = Q^2- in the immediate vicinity of a, since Q1 has no zeros. Two such equations cannot subsist simultaneously, except in one instance. Without entering into detailed discussion, the results obtained in the preceding chapters are sufficient to lead to an indication of the classification of singularities*. Singularities are said to be of the first class when they are accidental ; and a function is said to be of the first class when all its singularities are of the first class. It can, by § 48, have only a finite number of such singularities, each singularity being isolated. It is for this case alone that the two characteristic functions are in accord. When a function, otherwise of the first class, fails to satisfy the last condition, solely owing to failure of finiteness of multiplicity at some point, say at z = x , then that point ceases to be an accidental singularity. It has been called (§ 32) an essential singularity ; it belongs to the simplest kind of essential singularity ; and it is called a singularity of the second class. A function is said to be of the second class when it has some singularities of the second class ; it may possess singularities of the first class. By an argument similar to that adopted in § 48, a function of the second class can have only a limited number of singularities of the second class, each singularity being isolated. When a function, otherwise of the second class, fails to satisfy the last condition solely owing to unlimited condensation at some point, say at z = oo , of singularities of the second class, that point ceases to be a singularity of the second class: it is called a singularity (necessarily essential) of the third class. * For a detailed discussion, reference should be made to Guichard, " Theorie des points singnhers essentiels" (These, Gauthier-ViUars, Paris, 1883), who gives adequate references to the :stigations of Mittag-Leffler in the introduction of the classification and to the researches of Cantor. See also Mittag-Leffler, Acta Math., t. iv, (1884), pp. 1_79; Cantor Crelle t Ixxxiv 1878), pp. 242—258, Acta Math., t. ii, (1883), pp. 311—328. 10—2 148 CLASSIFICATION OF SINGULARITIES [88. A function is said to be of the third class when it has some singularities of the third class ; it may possess singularities of the first and the second classes. But it can have only a limited number of singularities of the third class, each singularity being isolated. Proceeding in this gradual sequence, we obtain an unlimited number of classes of singularities: and functions of the various classes can be constructed by means of the theorems which have been proved. A function of class n has a limited number of singularities of class n, each singularity being isolated, and any number of singularities of lower classes which, except in so far as they are absorbed in the singularities of class n, are isolated points. The effective limit of this sequence of classes is attained when the number of the class increases beyond any integer, however large. When once such a limit is attained, we have functions with essential singularities of unlimited class, each singularity being isolated ; when we pass to functions which have their essential singularities no longer isolated but, as in previous class-developments, of infinite condensation, it is necessary to add to the arrangement in classes an arrangement in a wider group, say, in species*. Calling, then, all the preceding classes of functions functions of the first species, we may, after Guichard (I.e.), construct, by the theorems already proved, a function which has at the points al} a*,... singularities of classes 1, 2,..., both series being continued to infinity. Such a function is called a function of the second species. By a combination of classes in species, this arrangement can be continued indefinitely ; each species will contain an infinitely increasing number of classes; and when an unlimited number of species is ultimately obtained, another wider group must be introduced. This gradual construction, relative to essential singularities, can be carried out without limit ; the singularities are the characteristics of the functions. * Guichard (I.e.) uses the term genre. CHAPTER VIII. MULTIFORM FUNCTIONS. 89. HAVING now discussed some of the more important general properties of uniform functions, we proceed to discuss some of the properties of multiform functions. Deviations from uniformity in character may arise through various causes : the most common is the existence of those points in the ^-plane, which have already (§ 12) been defined as branch-points. As an example, consider the two power-series Wl = l-i/-i/2-... , W2 = _(i_i/_^_... )f which, for points in the plane such that z' is less than unity, are the two values of (1 - /)* ; they may be regarded as two branches of the function w defined by the equation w2 = 1 — z' = z. Let / describe a small curve (say a circle of radius r) round the point z' = l, beginning on the axis of x\ the point 1 is the origin for z. Then z is r initially, and at the end of the first description of the circle z is re2wi ; hence initially wl is + 14 and w.2 is - r*} and at the end of the description w1 is -f r^e™ and w2 is — r^e™, that is, wl is — rf and w.2 is + ri Thus the effect of the single circuit is to change wl into w.2 and w2 into w1} that is, the effect of a circuit round the point, at which w1 and w2 coincide in value, is to interchange the values of the two branches. If, however, z describe a circuit which does not include the branch-point, wl and w2 return each to its initial value. Instances have already occurred, e.g. integrals of uniform functions, in which a variation in the path of the variable has made a difference in the 150 CONTINUATIONS [89. result; but this interchange of value is distinct from any of the effects produced by points belonging to the families of critical points which have been considered. The critical point is of a new nature ; it is, in fact, a characteristic of multiform functions at certain associated points. We now proceed to indicate more generally the character of the relation of such points to functions affected by them. The method of constructing a monogenic analytic function, described in § 34, by forming all the continuations of a power-series, regarded as a given initial element of the function, leads to the aggregate of the elements of the function and determines its region of continuity. When the process of con tinuation has been completely carried out, two distinct cases may occur. In the first case, the function is such that any and every path, leading from one point a to another point z by the construction of a series of successive domains of points along the path, gives a single value at z as the continuation of one initial value at a. When, therefore, there is only a single value of the function at a, the process of continuation leads to only a single value of the function at any other point in the plane. The function is uniform throughout its region of continuity. The detailed properties of such functions have been considered in the preceding chapters. In the second case, the function is such that different paths, leading from a to z, do not give a single value at z as the continuation of one and the same initial value at a. There are different sets of elements of the function, associated with different sets of consecutive domains of points on paths from a to z, which lead to different values of the function at z; but any change in a path from a to z does not necessarily cause a change in the value of the function at z. The function is multiform in its region of continuity. The detailed properties of such functions will now be considered. 90. In order that the process of continuation may be completely carried out, continuations must be effected, beginning at the domain of any point a and proceeding to the domain of any other point b by all possible paths in the region of continuity, and they must be effected for all points a and b. Continuations must be effected, beginning in the domain of every point a and returning to that domain by all possible closed paths in the region of continuity. When they are effected from the domain of one point a to that of another point b, all the values at any point z in the domain of a (and not merely a single value at such points) must be continued : and similarly when they are effected, beginning in the domain of a and returning to that domain. The complete region of the plane will then be obtained in which the function can be represented by a series of positive integral powers : and the boundary of that region will be indicated. 90.] OF A MULTIFORM FUNCTION 151 In the first instance, let the boundary of the region be constituted by a number, either finite or infinite, of isolated points, say L1} L2, Ls, ... Take any point A in the region, so that its distance from any of the points L is not infinitesimal ; and in the region draw a closed path ABC...EFA so as to enclose one point, say Ll} but only one point, of the boundary and to have no point of the curve at a merely infinitesimal distance from L^ Let such curves be drawn, beginning and ending at A, so that each of them encloses one and only one of the points of the boundary : and let Kr be the curve which encloses the point Lr. Let Wj be one of the power-series defining the function in a domain with its centre at A : let this series be continued along each of the curves Ks by successive domains of points along the curve returning to A. The result of the description of all the curves will be that the series w^ cannot be reproduced at A for all the curves though it may be reproduced for some of them ; otherwise, w: would be a uniform function. Suppose that w.2, w3> ..., each in the form of a power-series, are the aggregate of new distinct values thus obtained at A ; let the same process be effected on w2, w3, ... as has been effected on w1; and let it further be effected on any new distinct values obtained at A through w2, w3, ... , and so on. When the process has been carried out so far that all values obtained at A, by continuing any series round any of the curves K back to A, are included in values already obtained, the aggregate of the values of the function at A is complete : they are the values at A of the branches of the function. We shall now assume that the number of values thus obtained is finite, say n, so that the function has n branches at A : if their values be denoted by w1} w2, ..., wn, these n quantities are all the values of the function at A. Moreover, n is the same for all points in the plane, as may be seen by con tinuing the series at A to any other point and taking account of the corollaries at the end of the present section. The boundary-points L may be of two kinds. It may (and not infre quently does) happen that a point Ls is such that, whatever branch is taken at A as the initial value for the description of the circuit Ks, that branch is reproduced at the end of the circuit. Let the aggregate of such points be /u J2, .... Then each of the remaining points L is such that a description of the circuit round it effects a change on at least one of the branches, taken as an initial value for the description ; let the aggregate of these points be Blt 52, .... They are the branch-points; their association with the definition in § 12 will be made later. 152 DEFORMATION OF PATH [90. Fig. 15. When account is taken of the continuations of the function from a point A to another point B, we have n values at B as the continuations of n values at A. The selection of the individual branch at B, which is the continuation of a particular branch at A, depends upon the path of z between A and 5; it is governed by the following fundamental proposition : — The final value of a branch of a function for two paths of variation of the independent variable from one point to another will be the same, if one path can be deformed into the other without passing over a branch-point. Let the initial and the final points be a and b, and let one path of variation be acb. Let another path of variation be aeb, both paths lying in the region in which the function can be expressed by series of positive integral powers : the two paths are assumed to have no point within an infinitesimal distance of any of the boundary-points L and to be taken so close together that the circles of convergence of pairs of points (such as cx and e1} c2 and e2, and so on) along the two paths have common areas. When we begin at a with a branch of the function, values at d and at e^ are obtained, depending upon the values of the branch and its derivatives at a and upon the positions of ca and e^ hence, at any point in the area common to the circles of convergence of these two points, only a single value arises as derived through the initial value at a. Proceeding in this way, only a single value is obtained at any point in an area common to the circles of con vergence of points in the two paths. Hence ultimately one and the same value will be obtained at b as the continuation of the value of the one branch at a by the two different paths of variation which have been taken so that no boundary-point L lies between them or infinitesimally near to them. Now consider any two paths from a to b, say acb and adb, such that neither of them is near a boundary-point and that the contour they constitute does not enclose a boundary-point. Then by a series of successive infinitesimal deformations we can change the path acb to adb ; and as at b the same value of w is obtained for variations of z from a to b along the successive deformations, it follows that the same value of w is obtained at b for variations of z along acb as for varia tions along adb. Next, let there be two paths acb, adb constituting a closed contour, enclosing one (but not more than one) of the points / and none of the points B. When the original curve K which contains the point / is described, the initial value is restored : and hence the branches of the function obtained at any point of K by the two paths from any point, taken as initial point, are the same. By what precedes, the parts of this curve K can be deformed Fig. 16. 90.] OF THE VARIABLE 153 into the parts of acbda without affecting the branches of the function : hence the value obtained at b, by continuation along acb, is the same as the value there obtained by continuation along adb. It therefore follows that a path between two points a and b can be deformed over any point / without affecting the value of the function at b ; so that, when the preceding results are combined, the proposition enunciated is proved. By the continued application of the theorem, we are led to the following results : — COROLLARY I. Whatever be the effect of the description of a circuit on the initial value of a function, a reversal of the circuit restores the original value of the function. For the circuit, when described positively and negatively, may be re garded as the contour of an area of infinitesimal breadth, which encloses no branch-point within itself and the description of the contour of which therefore restores the initial value of the function. COROLLARY II. A circuit can be deformed into any other circuit without affecting the final value of the function, provided that no branch-point be crossed in tJie process of deformation. It is thus justifiable, and it is often convenient, to deform a path con taining a single branch-point into a loop round the point. A loop* consists of a line nearly to the point, °~ nearly the whole of a very small circle round the point, Fig. 17. and a line back to the initial point; see figure 17. COROLLARY III. The value of a function is unchanged when the variable describes a closed circuit containing no branch-point ; it is likewise unchanged when the variable describes a closed circuit containing all the branch-points. The first part is at once proved by remarking that, without altering the value of the function, the circuit can be deformed into a point. For the second part, the simplest plan is to represent the variable on Neumann's sphere. The circuit is then a curve on the sphere enclosing all the branch-points : the effect on the value of the function is unaltered by any deformation of this curve which does make it cross a branch-point. The curve can, without crossing a branch-point, be deformed into a point in that other part of the area of the sphere which contains none of the branch points ; and the point, which is the limit of the curve, is not a branch point. At such a point, the value of the function is unaltered ; and there fore the description of a circuit, which encloses all the branch-points, restores the initial value of the function. COROLLARY IV. If the values of w at b for variations along two paths * French writers use the word lacet, German writers the word Schleife. 154 EFFECT OF DEFORMATION [90. acb, adb be not the same, then a description of acbda will not restore the initial value of w at a. In particular, let the path be the loop OeceO (fig. 17), and let it change w at 0 into w'. Since the values of w at 0 are different and because there is no branch-point in Oe (or in the evanescent circuit OeO), the values of w at e cannot be the same : that is, the value with which the infinitesimal circle round a begins to be described is changed by the description of that circle. Hence the part of the loop that is effective for the change in the value of w is the small circle round the point ; and it is because the description of a small circle changes the value of w that the value of w is changed at 0 after the description of a loop. If/0?) be the value of w which is changed mtof^z) by the description of the loop, so that/Oz) and f^(z) are the values at 0, then the foregoing explanation shews that /(e) and / (e) are the values at e, the branch /(e) being changed by the description of the circle into the branch /i(e). From this result the inference can be derived that the points Bl} B.2, ... are branch-points as defined in § 12. Let a be any one of the points, and let f(z) be the value of w which is changed into f, (z) by the description of a very small circle round a. Then as the branch of w is monogenic, the difference between f(z) and f^(z) is an infinitesimal quantity of the same order as the length of the circumference of the circle : so that, as the circle is infinitesimal and ultimately evanescent, \f(z) -/iOz)| can be made as small as we please with decrease of z - a or, in the limit, the values of /(a) and /(a) at the branch-point are equal. Hence each of the points B is such that two or more branches of the function have the same value at the point and there is interchange among these branches when the variable describes a small circuit round the point : which affords a definition of a branch-point, more complete than that given in § 12. COROLLARY V. If a closed circuit contain several branch-points, the effect which it produces can be obtained by a combination of the effects produced in succession by a set of loops each going round only one of the branch-points. If the circuit contain several branch-points, say three as at a, b, c, then a path such as AEFD, in fig. 18, can without crossing any branch-point, be deformed into the loops AaB, BbC, GcD; and therefore the complete circuit AEFD A can be deformed validly into AaBbCcDA, and the same effect will be produced by the two forms of circuit. When D is made DA practically to coincide with A, the whole of the Fig. 18. second circuit is composed of the three loops. Hence the corollary. This corollary is of especial importance in the consideration of integrals of multiform functions. 91.] OF PATH OF THE VAKIABLE 155 COROLLARY VI. In a continuous part of the plane where there are no branch-points, each branch of a multiform function is uniform. Each branch is monogenic and, except at isolated points, continuous; hence, in such regions of the plane, all the propositions which have been proved for monogenic analytic functions can be applied to each of the branches of a multiform function. 91. If there be a branch-point within the circuit, then the value of the function at 6 consequent on variations along acb may, but will not necessarily, differ from its value at the same point consequent on variations along adb. Should the values be different, then the description of the whole curve acbda will lead at a not to the initial value of w, but to a different value. The test as to whether such a change is effected by the description is immediately derivable from the foregoing proposition; and as in Corollary IV., § 90, it is proved that the value is or is not changed by the loop, according as the value of w for a point near the circle of the loop is or is not changed by the description of that circle. Hence it follows that, if there be a branch-point which affects the branch of the function, a path of variation of the independent variable cannot be deformed across the branch point without a change in the value of w at the extremity of the path. And it is evident that a point can be regarded as a branch-point for a function only if a circuit round the point interchange some (or all) of the branches of the function which are equal at the point. It is not necessary that all the branches of the function should be thus affected by the point : it is sufficient that some should be interchanged*. Further, the change in the value of w for a single description of a circuit enclosing a branch-point is unique. For, if a circuit could change w into w' or w", then, beginning with w" and describing it in the negative sense we should return to w and afterwards describing it in the positive sense with w as the initial value we should obtain w'. Hence the circuit, described and then reversed, does not restore the original value w" but gives a different branch w' ; and no point on the circuit is a branch-point. This result is in opposition to Corollary I., of § 90 ; and therefore the hypothesis of alternative values at the end of the circuit is not valid, that is, the change for a single description is unique. But repetitions of the circuit may, of course, give different values at the end of successive descriptions. * In what precedes, certain points were considered which were regular singularities (see p. 163, note) and certain which were branch-points. Frequently points will occur which are at once branch-points and infinities ; proper account must of course be taken of them. 156 LAW OF INTERCHANGE [92. Fig. 19. 92. Let 0 be any ordinary point of the function ; join it to all the branch-points (generally assumed finite in number) in succession by lines which do not meet each other : then each branch is uniform for each path of variation of the variable which meets none of these lines. The effects pro duced by the various branch-points and their relations on the various branches can be indi cated by describing curves, each of which begins at a point indefinitely near 0 and returns to another point indefinitely near it after passing round one of the branch- points, and by noting the value of each branch of the function after each of these curves has been described. The law of interchange of branches of a function after description of a circuit round a branch-point is as follows: — All the branches of a function, which are affected by a branch-point as such, can either be arranged so that the order of interchange (for description of a path round the point) is cyclical, or be divided into sets in each of which the order of interchange is cyclical. Let wlt w.2> w3)... be the branches of a function for values of z near a branch-point a which are affected by the description of a small closed curve C round a : they are not necessarily all the branches of the function, but only those affected by the branch-point. The branch w^ is changed after a description of C ; let w2 be the branch into which it is changed. Then w2 cannot be unchanged by C; for a reversed description of C, which ought to restore w1} would otherwise leave w.2 un changed. Hence w2 is changed after a description of (7; it may be changed either into w1 or into a new branch, say w3. If into wlt then w-^ and w2 form a cyclical set. If the change be into w3, then w3 cannot remain unchanged after a description of C, for reasons similar to those that before applied to the change of w.2: and it cannot be changed into w2, for then a reversed de scription of G would change wz into w.A, and it ought to change w2 into w^ Hence, after a description of C, w3 is changed either into w^ or into a new branch, say w4. If into w1} then w1} w2, w3 form a cyclical set. If the change be into w4, then w4 cannot remain unchanged after a description of G ; and it cannot be changed into w.2 or ws, for by a reversal of the circuit that earlier branch would be changed into w4 whereas it ought to be changed into the branch, which gave rise to it by the forward descrip tion — a branch which is not w4. Hence, after a description of C, w4 is changed either into w^ or into a new branch. If into wlf then wj} w.2, w3, w4 form a cyclical set. 92.] OF BRANCHES OF A FUNCTION 157 If w4 be changed into a new branch, we proceed as before with that new branch and either complete a cyclical set or add one more to the set. By repetition of the process, we complete a cyclical set sooner or later. If all the branches be included, then evidently their complete system taken in the order in which they come in the foregoing investigation is a system in which the interchange is cyclical. If only some of the branches be included, the remark applies to the set constituted by them. We then begin with one of the branches not included in that set and evidently not inclusible in it, and proceed as at first, until we complete another set which may include all the remaining branches or only some of them. In the latter case, we begin again with a new branch and repeat the process ; and so on, until ultimately all the branches are included. The whole system is then arranged in sets, in each of which the order of interchange is cyclical. 93. The analytical test of a branch-point is easily obtained by con structing the general expression for the branches of a function which are interchanged there. Let z = a be a branch-point where n branches w1} ^v2,..., wn are cyclically interchanged. Since by a first description of a small curve round a, the branch w1 changes into w2, the branch w» into ws, and so on, it follows that by r descriptions w1 is changed into wr+l and by n descriptions wl reverts to its initial value. Similarly for each of the branches. Hence each branch returns to its initial value after n descriptions of a circuit round a branch point where n branches of the function are interchangeable. Now let z - a = Zn ; then, when z describes circles round a, Z moves in a circular arc round its origin. For each circumference described by z, the variable Z describes -th part of its circumference; and the complete circle is described by Z round its origin when n complete circles are described by z round a. Now the substitution changes wr as a function of z into a function of Z, say into Wr; and, after n complete descriptions of the ^-circle round a, wr returns to its initial value. Hence, after the description of a ^-circle round its origin, Wr returns to its initial value, that is, Z = 0 ceases to be a branch point for Wr. Similarly for all the branches W. But no other condition has been associated with a as a point for the function w ; and therefore Z = 0 may be any point for the function W, that is, it may be an ordinary point, or a singularity. In every case we have W a uniform function of Z in the immediate vicinity of the origin ; and therefore in that vicinity it can be expressed in the form 158 ANALYTICAL EXPRESSION [93. with the significations of P and G already adopted. When Z is an ordinary point, G is a constant or zero ; when Z is an accidental singularity, O is an algebraical function ; and, when Z is an essential singularity, G is a transcen dental function. The simpler cases are, of course, those in which the form of G is alge braical or constant or zero ; and then W can be put into the form ZmP(Z), where P is an infinite series of positive powers and m is an integer. As this is the form of W in the vicinity of Z=Q, it follows that the form of w in the vicinity of z = a is m 1 (z - a)n P {(z - a)n} and the various n branches of the function are easily seen to be given by i substituting in the above for (z — a)n the values 2im j. e m (z — of, where s = 0, 1,..., n — 1. We therefore infer that the general expression for the n branches of a function, which are interchanged by circuits round a branch-point z = a, assumed not to be an essential singularity, is m _ 1 (z - a)Tl P {(z - a)»}, i where m is an integer, and where to (z — a)n its n values are in turn assigned to obtain the different branches of the function. There may be, however, more than one cyclical set of branches. If there be another set of r branches, then it may similarly be proved that their general expression is OTI _ i (zjaYQ{(z-ay-}, where m^ is an integer, and Q is an integral function ; the various branches i are obtained by assigning to (z — a)r its r values in turn. And so on, for each of the sets, the members of which are cyclically interchangeable at the branch-point. When the branch-point is at infinity, a different form is obtained. Thus in the case of a set of n cyclically interchangeable branches we take z = %-», so that n negative descriptions of a closed £-curve, excluding infinity and no other branch-point, requires a single positive description of a closed curve round the w-origin. These n descriptions restore the value of w; as a function of z to its initial value; and therefore the single description of the M- curve round the origin restores the value of U — the equivalent of w after the 93.] NEAR A BRANCH-POINT 159 change of the independent variable — as a function of u. Thus u = 0 ceases to be a branch-point for the function U ; and therefore the form of U is . . where the symbols have the same general signification as before. If, in particular, z = oo be a branch-point but not an essential singularity, then G is either a constant or an algebraical function ; and then U can be expressed in the form u~mP(u}, where TO is an integer. When the variable is changed from u to z, then the general expression for the n branches of a function which are interchangeable at z = oo , assumed not to be an essential singularity, is where TO is an integer and where to zn its n values are assigned to obtain the different branches of the function. If, however, the branch-point z = a in the former case or z = oo in the latter be an essential singularity, the forms of the expressions in the vicinity of the point are _i i G{(z-a) »J>-p{(jr-aJ»}f i _i and G(zn) + P(z n), respectively. Note. When a multiform function is denned, either explicitly or im plicitly, it is practically always necessary to consider the relations of the branches of the function for z = oo as well as their relations for points that are infinities of the function. The former can be determined by either of the processes suggested in § 4 for dealing with z=<x>; the latter can be determined as in the present article. Moreover, the total number of branches of the function has been assumed to be finite. The cases, in which the number of branches is unlimited, need not be discussed in general : it will be sufficient to consider them when they arise, as they do arise, e.g., when the function is of the form of an algebraical irrational with an irrational index such as z^ — hardly a function in the ordinary sense—, or when the function is the logarithm of a function of z, or is the inverse of a periodic function. In the nature of their multiplicity of branching and of their sequence of interchange, they are for the most part distinct from the multiform functions with only a finite number of branches. Ex. The simplest illustrations of multiform functions are furnished by functions denned by algebraical equations, in particular, by algebraic irrationals. 160 ALGEBRAICAL [93. The general type of the algebraical irrational is the product of a number of functions of the form w = {A(z — al)(z-a.2) ...... (z-a^)}m, m and n being integers. This particular function has m branches; the points a1} «2, ...... , an are branch-points. To find the law of interchange, we take z-ar = pe01; then when a small circle of radius p is described round ar, so that z returns to its initial position, the value of 6 increases by 2n and the new value of w is aw, where a is the with root of unity defined by em m. Taking then the various branches as given by w, aw, a?w, ...... , am~lw, we have the law of inter change for description of a small curve round any one-branch point as given by this succession in cyclical order. The law of succession for a circuit enclosing more than one of the branch-points is derivable by means of Corollary V, § 90. To find the relation of z = o> to w, we take zz' = l and consider the new function W in the vicinity of the ^'-origin. We have W ={A (1 -VH1 -a/) ...... (l-an<)}^'~»». If the variable z1 describe a very small circle round the origin in the negative sense, then 27TZ — z' is multiplied by e~2™ and so W acquires a factor e ™, that is, W is changed unless this acquired factor is unity. It can be unity only when n/m is an integer ; and therefore, except when n/m is an integer, 0=00 is a branch-point of the function. The law of succession is the same as that for negative description of the z'-circle, viz., w, anw, a2nw, ...... ; the m values form a single cycle only if n be prime to m, and a set of cycles if n be not prime to m. Thus 0=00 is a branch -point for w = (k?-gg-g^~^ ; it is not a branch-point for w = {(\ -22) (1 — &2z2)}~*; and z = b is a branch-point for the function defined by (z — b) w2 = z — a, but z = b is riot a branch-point for the function defined by (z—b)2wz = z-a. Again, if p denote a particular value of ft when z has a given value, and q similarly denote a particular value of [— — : ) , then w=p+q is a six-valued function, the values V+v being W6= -p + aq, where a is a primitive cube root of unity. The branch-points are - 1, 0, 1, oo ; and the orders of change for small circuits round one (and only one) of these points are as follows : For a small circuit round -1 0 1 00 Wj changes to •ft W-i W3 W2 W2 „ we W1 W4 w, ^3 to, W4 W5 W4 W4 „ M>2 w, W6 W3 Ws n W3 W6 w, W6 U>6 W74 W5 Wo W5 93.] FUNCTIONS 161 Combinations can at once be effected ; thus, for a positive circuit enclosing both 1 and QO but* not — 1 or 0, the succession is iolt w4, w6, w2, w3, WG in cyclical order. 94. It has already been remarked that algebraic irrationals are a special class of functions denned by algebraical equations. Functions thus generally denned by equations, which are algebraical so far as concerns the dependent variable but need not be so in reference to the independent variable, are often called algebraical. The term, in one sense, cannot be strictly applied to the roots of an equation of every degree, seeing that the solution of equations of the fifth and higher degrees can be effected only by transcendental functions; but what is implied is that a finite number of determinations of the dependent variable is given by the equation -f*. The equation is algebraical in relation to the dependent variable w, that is, it will be taken to be of finite degree n in w. The coefficients of the different powers will be supposed to be rational uniform functions of z : were they irrational in any given equation, the equation could be transformed into another, the coefficients of which are rational uniform functions. And the equation is supposed to be irreducible, that is, if the equation be taken in the form f(w, *) = 0, the left-hand member f(w, z) cannot be resolved into factors of a form and character as regards w and z similar to /itself. The existence of equal roots of the equation for general values of z requires that fi \ j "df(w> z) f(w,z) and ^~ shall have a common factor, which will be rational owing to the form of f(w, z}. This form of factor is excluded by the irreducibility of the equation ; so that /= 0, as an equation in w, has not equal roots for general values of z. But though the two equations are not both satisfied in virtue of a simpler equation, they are two equations determining values of w and #; and their form is such that they will give equal values of w for special values of z. Since the equation is of degree n, it may be taken to be w where the functions F1} F2}... are rational and uniform. If all their singu- * Such a circuit, if drawn on the Neumann's sphere, may be regarded as excluding - 1 and 0, or taking account of the other portion of the surface of the sphere, it may be regarded as a negative circuit including - 1 and 0, the cyclical interchange for which is easily proved to be iCj, w4, w5, w.2, M?3, w6 as in the text. t Such a function is called Men defini by Liouville. F. 11 162 ALGEBRAICAL [94. larities be accidental, they are raeromorphic algebraical functions of z (unless z = oo is the only singularity, in which case they are holomorphic) ; and the equation can then be replaced by one which is equivalent and has all its coefficients holomorphic, the coefficient of wn being the least common multiple of all the denominators of the meromorphic functions in the first form. This form cannot however be deduced, if any of the singularities be essential. The equation, as an equation in w, has n roots, all functions of z ; let these be denoted by w1,w2,..., ivn, which are the n branches of the function w. When the geometrical interpretation is associated with the analytical relation, there are n points in the w-plane, say a1,..., an, which correspond with a point in the ^-plane, say with c^ ; and in general these n points are distinct. As z varies so as to move in its own plane from a, then each of the w-points moves in their common plane ; and thus there are n w-paths corresponding to a given z-path. These n curves may or may not meet one another. If they do not, there are n distinct w-paths, leading from a1;..., an to /3i,..., /3n, respectively corresponding to the single ^-path leading from a to b. If two or more of the w-paths do meet one another, and if the describing w-poirits coincide at their point of intersection, then at such a point of intersection in the w-plane, the associated branches w are equal ; and therefore the point in the ^-plane is a point that gives equal values for w. It is one of the roots of the equation obtained by the elimination of w between the analytical test as to whether the point is a branch-point will be considered later. The march of the concurrent ^-branches from such a point of intersection of two w-paths depends upon their relations in its immediate vicinity. When no such point lies on a ^-path from a to b, no two of the w-points coincide during the description of their paths. By § 90, the 2-path can be deformed (provided that, in the deformation, it does not cross a branch-point) without causing any two of the w-points to coincide. Further, if z describe a closed curve which includes none of the branch -points, then each of the ^-branches describes a closed curve and no two of the tracing points ever coincide. Note. The limitation for a branch-point, that the tracing w-points coincide at the point of intersection of the w-curves, is of essential im portance. What is required to establish a point in the z-plane as a branch-point, is not a mere geometrical intersection of a couple of completed w-paths but the coincidence of the w-points as those paths are traced, together with inter- 94.] FUNCTIONS 1 63 change of the branches for a small circuit round the point. Thus let there be such a geometrical intersection of two w-curves, without coincidence of the tracing points. There are two points in the ^-plane corresponding to the geometrical intersection ; one belongs to the intersection as a point of the w-paih which first passed through it, and the other to the intersection as a point of the w-path which was the second to pass through it. The two branches of w for the respective values of z are undoubtedly equal ; but the equality would not be for the same value of z. And unless the equality of branches subsists for the same value of z, the point is not a branch point. A simple example will serve to illustrate these remarks. Let w be defined by the equation so that the branches w1 and w2 are given by Ci0j_ = cz+z(z2 + c2)*, cw2 = cz-z(z*-\- c2)* ; it is easy to prove that the equation resulting from the elimination of w between /=0 and and that only the two points z= ±ic are branch-points. The values of z which make wl equal to the value of wz for z = a (supposed not equal to either 0, ci or — ci) are given by cz + z (02 + c2)* = ca - a (a2 + c2)*, which evidently has not 2 = a for a root. Rationalising the equation so far as concerns z and removing the factor z -a, as it has just been seen not to furnish a root, we find that s is determined by z3 + z2a + za2 + a3 + 2ac2 - 2ac (a2 + c2) * = 0, the three roots of which are distinct from a, the assumed point, and from ±ci, the branch point. Each of these three values of z will make wv equal to the value of w2 for z=a : we have geometrical intersection without coincidence of the tracing points. 95. When the characteristics of a function are required, the most im portant class are its infinities: these must therefore now be investigated. It is preferable to obtain the infinities of the function rather than the singularities alone, in the vicinity of which each branch of the function is uniform * : for the former will include these singularities as well as those branch-points which, giving infinite values, lead to regular singularities when the variables are transformed as in § 93. The theorem which deter mines them is: — The infinities of a function determined by an algebraical equation are the singularities of the coefficients of the equation. Let the equation be wn + wn-i FI ^ + wn-, !»,(*) + ... + rf^ (^) + ^ (^) = Q, * These singularities will, for the sake of brevity, be called regular. 11—2 164 INFINITIES [95. and let w' be any branch of the function; then, if the equation which determines the remaining branches be wn-i + wn-2 Qi ^ + wn-3 £2 (Y) + . . . + WGn-z (Z) + Gn-i (z) = 0, we have Fn (z) = - w'Gn-i (z), Fn^ (z) = - w'Gn-z (z) + £„_! (z), ^71-2 (Z) = - w'Gn-s (z} + #n-2 (z), Now suppose that a is an infinity of w' ; then, unless it be a zero of order at least equal to that of Gn^ (z), a is an infinity of Fn (z). If, however, it be a zero of Gn-i (z) of sufficient order, then from the second equation it is an infinity of Fn_l(z) unless it is a zero of order at least equal to that of 6rn_2 (z) ; and so on. The infinity must be an infinity of some coefficient not earlier than Fi (z) in the equation, or it must be a zero of all the functions G which are later than Gf_! (z). If it be a zero of all the functions Gr, so that we may not, without knowing the order, assert that it is of rank at least equal to its order as an infinity of w', still from the last equation it follows that a must be an infinity of Fl (z). Hence any infinity of w is an infinity of at least one of the coefficients of the equation. Conversely, from the same equations it follows that a singularity of one of the coefficients is an infinity either of w' or of at least one of the co efficients G. Similarly the last alternative leads to an inference that the infinity is either an infinity of another branch w" or of the coefficients of the (theoretical) equation which survives when the two branches have been removed. Proceeding in this way, we ultimately find that the infinity either is an infinity of one of the branches or is an infinity of the coefficient in the last equation, that is, of the last of the branches. Hence any singularity of a coefficient is an infinity of at least one of the branches of the function. It thus appears that all the infinities of the function are included among, and include, all the singularities of the coefficients ; but the order of the infinity for a branch does not necessarily make that point a regular singularity nor, if it be made a regular singularity, is the order necessarily the same as for the coefficient. 96. The following method is effective for the determination of the order of the infinity of the branch. Let a be an accidental singularity of one or more of the F functions, say of order ra; for the function Ft ; and assume that, in the vicinity of a, we have Ft (z) = (z- a)-™* [Ci + di (z-a) + e{ (z - a? +...]. 96.] OF ALGEBRAICAL FUNCTIONS 165 Then the equation which determines the first term of the expansion of w in a series in the vicinity of a is wn + d (z — a)~™i wn~l + c2(z — a)~m2 wn~2 + ... -f cn_! (z - a)~m»-i w + cn (z - a)~m« = 0. Mark in a plane, referred to two rectangular axes, points n, 0; n — 1, — m^; n — 2, — m2 ; . . ., 0, — mn ; let these be A0, A1} ..., An respectively. Any line through Ai has its equation of the form 1 1 —I— nm • ~~ ~\ J o" — l w — T. t)\\ y T »*< — A, {J, (71, tftf that is, y — \x = — \ (n — i) — mi. If then w = (z — a)~xf(z}, where f(z) is finite when z = a, the intercept of the fore going line on the negative side of the axis of y is equal to the order of the infinity in the term wn-iFi(z). This being so, we take a line through An coinciding in direction with the negative part of the axis of y and we turn it about An in a trigonometrically positive direction until it first meets one of the other points, say An_r ; then we turn it about An_r until it meets one of the other points, say An_s; and so on until it passes through A0. There will thus be a line from An to A0, generally consisting of a number of parts ; and none of the points A will be outside it. The perpendicular from the origin on the line through An_r and An_g is evidently greater than the perpendicular on any parallel line through a point A, that is, on any line through a point A with the same value of X; and, as this perpendicular is it follows the order of the infinite terms in the equation, when the particular substitution is made for w, is greater for terms corresponding to points lying on the line than it is for any other terms. If /(*) = 0 wnen z = a, then the terms of lowest order after the substitu tion of (z — a)~Kf(z) for w are as many terms occurring in the bracket as there are points A on the line joining An_r to An_s. Since the equation determining w must be satisfied, terms of all orders must disappear, and therefore an equation determining s-r values of 6, that is, the first terms in the expansions of s — r branches w. 166 INFINITIES [96. Similarly for each part of the line : for the first part, there are r branches with an associated value of X ; for the second, s — r branches with another associated value ; for the third, t — s branches with a third associated value ; and so on. The order of the infinity for the branches is measured by the tangent of the angle which the corresponding part of the broken line makes with the axis of a; ; thus for the line joining An^. to An_s the order of the infinity for the s - r branches is where mn_r and mn_s are the orders of the accidental singularities of Fn_r (z) and Fn_s (z). If any part of the broken line should have its inclination to the axis of x greater than \ir so that the tangent is negative and equal to - //,, then the form of the corresponding set of branches w is (z — a,y g {z} for all of them, that is, the point is not an infinity for those branches. But when the inclination of a part of the line to the axis is < \TT, so that the tangent is positive and equal to X, then the form of the corresponding set of branches w is (z — a)~Kf(z) for all of them, that is, the point is an infinity of order X for those branches. In passing from An to A0 there may be parts of the broken line which have the tangential coordinate negative, implying therefore that a is not an infinity of the corresponding set or sets of branches w. But as the revolving line has to change its direction from Any' to some direction through A0, there must evidently be some part or parts of the broken line which have their tangential coordinate positive, implying therefore that a is an infinity of the corresponding set or sets of branches. Moreover, the point a is, by hypothesis, an accidental singularity of at least one of the coefficients and it has been supposed to be an essential singularity of none of them; hence the points A0, A1} ..., An are all in the finite part of the plane. And as no two of their abscissa are equal, no line joining two of them can be parallel to the axis of y, that is, the inclination of the broken line is never \ir and therefore the tangential coordinate is finite, that is, the order of the infinity for the branches is finite for any accidental singularity of the coefficients. If the singularity at a be essential for some of the coefficients, the corresponding result can be inferred by passing to the limit which is obtained by making the corresponding value or values of m infinite. In that case the corresponding points A move to infinity and then parts of the broken line pass through A0 (which is always on the axis of x) parallel to the axis of y, that is, the tangential coordinate is infinite and the order of 96.] OF ALGEBRAICAL FUNCTIONS 167 the infinity at a for the corresponding branches is also infinite. The point is then an essential singularity (and it may be also a branch-point). It has been assumed implicitly that the singularity is at a finite point in the 2-plane ; if, however, it be at oo , we can, by using the transformation zz' — 1 and discussing as above the function in vicinity of the origin, obtain the relation of the singularity to the various branches. We thus have the further proposition : The order of ike infinity of a branch of an algebraical function at a singularity of a coefficient of the equation, which determines the function, is finite or infinite according as the singularity is accidental or essential. If the coefficients FI of the equation be holomorphic functions, then z = oo is their only singularity and it is consequently the only infinity for branches of the function. If some of or all the coefficients Ff be mero- morphic functions, the singularities of the coefficients are the zeros of the denominators and, possibly, £=oo; and, if the functions be algebraical, all such singularities are accidental. In that case, the equation can be modified to h0 (z) wn + h^ (z} wn~l + A2 (z) wn~2 + . . . = 0, where h0(z) is the least common multiple of all the denominators of the functions Ft. The preceding results therefore lead to the more limited theorem : When a function w is determined by an algebraical equation the coefficients of which are holomorphic functions of z, then each of the zeros of the coefficient of the highest power of w is an infinity of some of (and it may be of all) the branches of the function w, each such infinity being of finite order. The point z= oo may also be an infinity of the function w ; the order of that infinity is finite or infinite according as z = oo is an accidental or an essential singularity of any of the coefficients. It will be noticed that no precise determination of the forms of the branches w at an infinity has been made. The determination has, however, only been deferred : the infinities of the branches for a singularity of the coefficients are usually associated with a branch-point of the function and therefore the relations of the branches at such a point will be of a general character independent of the fact that the point is an infinity. If, however, in any case a singularity of a coefficient should prove to be, not a branch-point of w but only a regular singularity, then in the vicinity of that point the branch of w is a uniform function. A necessary (but not suffi cient) condition for uniformity is that (mn_r — mn_s) -7- (s — r) be an integer. Note. The preceding method can be applied to determine the leading terms of the branches in the vicinity of a point a which is an ordinary point for each of the coefficients F. 168 BRANCH-POINTS [97. 97. There remains therefore the consideration of the branch-points of a function determined by an algebraical equation. The characteristic property of a branch-point is the equality of branches of the function for the associated value of the variable, coupled with the interchange of some of (or all) the equal branches after description by the variable of a small contour enclosing the point. So far as concerns the first part, the general indication of the form of the values has already (§ 93) been given. The points, for which values of w determined as a function of z by the equation f(w, z) =J0 are equal, are determined by the solution of this equation treated simul taneously with df(w, z) = Q. dw and when a point z is thus determined the corresponding values of w, which are equal there, are obtained by substituting that value of z and taking M, the greatest common measure of / and -J- . The factors of M then lead to the value or the values of w at the point ; the index m of a linear factor gives at the point the multiplicity of the value which it determines, and shews that m + 1 values of w have a common value there, though they are distinct at infinitesimal distances from the point. If m = 1 for any factor, the corresponding value of w is an isolated value and determines a branch that is uniform at the point. Let z = a, w = a be a value of z and a value of w thus obtained ; and suppose that m is the number of values of w that are equal to one another. The point z = a is not a branch-point unless some interchange among the in values of w is effected by a small circuit round a ; and it is therefore necessary to investigate the values of the branches* in the vicinity of z — a. Let w = a. + w', z = a + z' ; then we have that is, on the supposition that f(w, z) has been freed from fractions, /(a, a) + SS^rXV = 0, r, s so that, since a is a value of w corresponding to the value a of z, we have w' and / connected by the relation * The following investigations are founded on the researches of Puiseux on algebraic functions; they are contained in two memoirs, Liouville, lre Ser., t. xv, (1850), pp. 365 — 480, ib., t. xvi, (1851), pp. 228—240. See also the chapters on algebraic functions, pp. 19 — 76, in the second edition of Briot and Bouquet's Theorie des fonctions elliptiques. 97.] OF ALGEBRAICAL FUNCTIONS 169 When / is 0, the zero value of w' must occur m times, since a is a root m times repeated; hence there are terms in the foregoing equation inde pendent of z, and the term of lowest index among them is w'm. Also when w ' = 0, z' — 0 is a possible root ; hence there must be a term or terms independent of w' in the equation. First, suppose that the lowest power of z among the terms independent of w' is the first. The equation has the form Az' + higher powers of z' + Biu' + higher powers of w' + terms involving z' and w' = 0, O-/* / \ where A is the value of - ' — - for w = a, z = a. Let z'=%m, w' = v%: the 02 last form changes to (A + Bvm) £m + terms with £m+1 as a factor = 0 ; and therefore A + Bvm + terms involving £= 0. Hence in the immediate vicinity of z = a, that is, of £ =0, we have A + Bv™ = 0. Neither A nor B is zero, so that all the m values of v are finite. Let them be vl}..., vm, so arranged that their arguments increase by 2-Tr/Tn through the succession. The corresponding values of w' are for i = l, ..., m. Now a ^-circuit round a, that is, a /-circuit round its origin, increases the argument of z' by 2?r ; hence after such a circuit we 1_ 27Tt !_ have the new value of w{ as ViZ/m em, that is, it is vi+1z'm which is the value of w'i+l. Hence the set of values w\, «/.,,..., w'm form a complete set of interchangeable values in their cyclical succession ; all the m values, which are equal at a, form a single cycle and the point is a branch-point. Next, suppose that the lowest power of z among the terms independent of w is z' , where I > 1. The equation now has the form 0 = Az' + higher powers of z' + Bw' + higher powers of w' Arsz'V r=l s=l where in the last summation r and s are not zero and in every term either (i), r is equal to or greater than I or (ii), s is equal to or greater than m or (iii), both (i) and (ii) are satisfied. As only terms of the lowest orders 170 BRANCH-POINTS [97. need be retained for the present purpose, which is the derivation of the first term of w' in its expansion in powers of z', we may use the foregoing equation in the form , l-lm-l A/ + 2 2 A, r=l s=l ,r ,s , -p. ,m _ jf w + Bw = 0. To obtain this first term we proceed in a manner similar to that in § 96 *. Points A0,..., Am are taken in a plane referred to rectangular axes having as co ordinates 0, £;...; s, r;...; m, 0 respectively. A line is taken through Am and is made to turn round Am from the position AmO until it first meets one of the other points ; then round the last point which lies in this direction, say round Aj, until it first meets another ; and so on. Any line through At (the point si} rt) is of the form y - Ti = - \ (x - s^. The intercept on the axis of /-indices is \Si + Ti, that is, the order of the term involving Ars for a substitution w' oc / . The perpendicular from the origin for a line through AI and Aj is less than for any parallel line through other points with the same inclination ; and, as this perpendicular is Fig. 21. it follows that, for the particular substitution w' oc z' , the terms corresponding to the points lying on the line with coordinate X are the terms of lowest order and consequently they are the terms which give the initial terms for the associated set of quantities w'. Evidently, from the indices retained in the equation, the quantities X for the various pieces of the broken line from Am to A0 are positive and finite. Consider the first piece, from Am to Aj say ; then taking the value of X for that piece as fa, so that we write v^z'*1 as the first term of w', we have as the set of terms involving the lowest indices J? /"* i ^ ^ A fl* fi I A fl*i , ^J Sj being the smallest value of s retained ; and then so that /*! = m — s * Reference in this connection may be made to Chrystal's Algebra, ch. xxx., with great advantage, as well as the authorities quoted on p. 168, note. GROUPING OF BRANCHES 171 Let p/q be the equivalent value of ^ as the fraction in its lowest terms ; and p write / = (?. Then w' = vlz'i = vtf ; all the terms except the above group are of order > mp and therefore the equation leads after division by %mPtfi to Bv^-'i + ^Aravf-*i + Arfj = 0, an equation which determines m — Sj values for vl, and therefore the initial terms of m — Sj of the w-branches. Consider now the second piece, from Aj to At say ; then taking the value of A, for that piece as fa, so that we write v.2z'^ as the first term of w', we have as the set of terms involving the lowest indices for this value of /*2 A fri /s.- xr"O A iv Is fl"i tsi Arfz' Jw ' + E&A.rjt w' + Ar.sz' *w \ where S{ is the smallest value of s retained. Then Sjfr + Tj = tyig + r Proceeding exactly as before, we find as the equation determining Sj-Si values for v2 and therefore the initial terms of Sj — st of the w-branches. And so on, until all the pieces of the line are used ; the initial terms of all the w-branches are thus far determined in groups connected with the various pieces of the line A^Ai^.A,. By means of these initial terms, the m-branches can be arranged for their interchanges, by the description of a small circuit round the branch-point, according to the following theorem :— Each group can be resolved into systems, the members of each of which are cyclically interchangeable. It will be sufficient to prove this theorem for a single group, say the group determined by the first piece of broken line: the argument is general. Since - is the equivalent of — ^— and of T} . and since s, < s, we have V m — s m — Sj m-s = kq, m-Sj^kjq, kj>k; and then the equation which determines ^ is Sv&v + 2^r,,Vl <*,-*> 1 4 ArjSj = 0, that is, an equation of degree k} in vj as its variable. Let U be any root of it ; then the corresponding values of vl are the values of U*. Suppose these q values to be arranged so that the arguments increase by 27r^, which is possible, because p is prime to q. Then the q values of w' being the values of v^Vi are P. P P 172 GROUPING OF BRANCHES [97. where vla is that value of Ifi which has — — for its argument. A circuit round the /-origin evidently increases the argument of any one of these w'-values by Zrrp/q, that is, it changes it into the value next in the succession; and so the set of q values is a system the members of which are cyclically interchangeable. This holds for each value of U derived from the above equation ; so that the whole set of m — Sj branches are resolved into kj systems, each containing q members with the assigned properties. It is assumed that the above equation of order kj in vj has its roots unequal. If, however, it should have equal roots, it must be discussed ab initio by a method similar to that for the general equation; as the order kj (being a factor of m — Sj) is less than m, the discussion will be shorter and simpler, and will ultimately depend on equations with unequal roots as in the case above supposed. It may happen that some of the quantities /j, are integers, so that the corresponding integers q are unity : a number of the branches would then be uniform at the point. It thus appears that z = a is a branch-point and that, under the present circumstances, the m branches of the function can be arranged in systems, the members of each one of which are cyclically interchangeable. Lastly, it has been tacitly assumed in what precedes that the common value of w for the branch-point is finite. If it be infinite, this infinite value can, by § 95, arise only out of singularities of the coefficients of the equation : and there is therefore a reversion to the discussion of §§ 95, 96. The dis tribution of the various branches into cyclical systems can be carried out exactly as above. Another method of proceeding for these infinities would be to take ww' = \, z= c + z' ; but this method has no substantial advantage over the earlier one and, indeed, it is easy to see that there is no substantial difference between them. Ex. 1. As an example, consider the function determined by the equation The equation determining the values of z which give equal roots for w is 82 (2 -1)2 = 4(3 -I)3 so that the values are z=l (repeated) and z= — 1. When z=l, then w=0, occurring thrice; and, if 2 = 1+2' then 8W/3W, that is, w'^^z13. The three values are branches of one system in cyclical order for a circuit round z=\. 97.] EXAMPLES 173 When z = — 1, the equation for w is that is, (w so that w=\ or w= — £, occurring twice. For the former of these we easily find that, for s= — l-\-z', the value of w is l-hfs'-f ...... , an isolated branch as is to be expected, for the value 1 is not repeated. For the latter we take w—— \ + w' and find so that the two branches are and they are cyclically interchangeable for a small circuit round z= - 1. These are the finite values of w at branch-points. For the infinities of w, which may arise in connection with the singularities of the coefficients, we take the zeros of the coefficient of the highest power of w in the integral equation, viz., 2 = 0, which is thus the only infinity of w. To find its order we take w=z~nf (z)—yz~n + ...... , where y is a constant and f (z) is finite for 2 = 0; and then we have 8zl~3n J. "~ Thus l-3n=-n, provided both of them be negative; the equality gives n = \ and satisfies the condition. And 8y3= - 3y. Of these values one is zero, and gives a branch of the function without an infinity; the other two are ±^V-f and they give the initial term of the two branches of w, which have an infinity of order -^ at the origin and are cyclically interchangeable for a small circuit round it. The three values of w for infinitesimal values of z are 3 . _i 1 7 /3 . l 4 275 " - 81 '-1944 3 • - M + — /?&*— 1*4. 215 /3-f_jL2_ 6 18 V 8 81 1944 V 8 729 z _ _i A As w3--g + gj2+— 2 + The first two of these form the system for the branch-point at the origin, which is neither an infinity nor a critical point for the third branch of the function. Ex. 2. Obtain the branch-points of the functions which are defined by the following equations, and determine the cyclical systems at the branch-points : (i) w* (ii) w (iii) w (iv) iff 44 (v) vfi - (1 - a2) 104 _ _ Z2 (! _ 22)4 = 0- (Briot and Bouquet.) Also discuss the branches, in the vicinity of 2 = 0 and of 2=00, of the functions defined by the following equations : (vi) aw7 + bu£z + cutz* + dwW + ewz1 +fz9 + gv£ + hw*£ + kzw = 0 ; (vii) wmzn+wn+zm = Q. 174 SIMPLE BRANCH-POINTS [98. 98. There is one case of considerable importance which, though limited in character, is made the basis of Clebsch and Gordan's investigations* in the theory of Abelian functions — the results being, of course, restricted by the initial limitations. It is assumed that all the branch-points are simple, that is, are such that only one pair of branches of w are interchanged by a circuit of the variable round the point ; and it is assumed that the equation /= 0 is algebraical not merely in w but also in z. The equation f = 0 can then be regarded as the generalised form of the equation of a curve of the nth order, the generalisation consisting in replacing the usual coordinates by complex variables; and it is further assumed, in order to simplify the analysis, that all the multiple points on the curve are (real or imaginary) double-points. But, even with the limitations, the results are of great value : and it is therefore desirable to establish the results that belong to the present section of the subject. We assume, therefore, that the branch-points are such that only one pair of branches of w are interchanged by a small closed circuit round any one of the points. The branch-points are among the values of z determined by the equations z) A > When /=0 has the most general form consistent with the assigned limitations, f (w, z) is of the ?ith degree in z ; the values of z are determined by the eliminant of the two equations which is of degree n(n — 1), and there are, therefore, n(n — Y) values of z which must be examined. First, suppose that J \, ' — ' does not vanish for a value of z, thus oz obtained, and the corresponding value of w : then we have the first case in the preceding investigation. And, on the hypothesis adopted in the present instance, m = 2 ; so that each such point z is a branch-point. Next, suppose that — ^ - vanishes for some of the n(n — 1) values of z ; the value of m is still 2, owing to the hypothesis. The case will now be still d'2f (w z} further limited by assuming that ^ .2 does not vanish for the value of z and the corresponding value of w ; and thus in the vicinity of z = a, w = a we have an equation 0 = Az- + 2Bz'w' + Cw'2 -f terms of the third degree + ...... , where A, B, C are the values of ^ , =-£- , «~ f°r z — a> w=a. oz1 dzdw ow2 If B2 AC, this equation leads to the solution C'w + Bz oc uniform function of z. * Clebsch und Gordan, Theorie der AbeVschen Functionen, (Leipzig, Teubner, 1866). 98.] SIMPLE BRANCH-POINTS 175 The point z = a, w = a is not a branch-point ; the values of w, equal at the point, are functionally distinct. Moreover, such a point z occurs doubly in the eliminant; so that, if there be B such points, they account for 28 in the eliminant of degree n (n — 1) ; and therefore, on their score, the number n (n — 1) must be diminished by '28. The case is, reverting to the genera lisation of the geometry, that of a double point where the tangents are not coincident. If, however, B2 = AC, the equation leads to the solution Cw' + Bz' = Lz'^ + Mz'* + Nz'* + The point z = a, w = a is a point where the two values of z interchange. Now such a point z occurs triply in the eliminant ; so that, if there be K such points, they account for SK of the degree of the equation. Each of them provides only one branch-point, and the aggregate therefore provides K branch-points ; hence, in counting the branch-points of this type as derived through the eliminant, its degree must be diminished by 2/c. The case is, reverting to the generalisation of the geometry, that of a double point (real or imaginary) where the tangents are coincident. It is assumed that all the n(n— 1) points z are accounted for under the three classes considered. Hence the number of branch-points of the equation is £l = n (n - 1) - 28 - 2«, where n is the degree of the equation, B is the number of double points (in the generalised geometrical sense) at which tangents to the curve do not coincide, and K is the number of double points at which tangents to the curve do coincide. And at each of these branch-points, II in number, two branches of the function are equal and, for a small circuit round it, interchange. 99. The following theorem is a combined converse of many of the theorems which have been proved : A function w, which has n (and only ?/) values for each value of z, and which has a finite number of infinities and of branch-points in any part of the plane, is a root of an equation in w of degree n, the coefficients of which are uniform functions of z in that part of the plane. We shall first prove that every integral symmetric function of the n values is a uniform function in the part of the plane under consideration. n Let Sk denote 2, w£, where k is a positive integer. At an ordinary point i-\ of the plane, Sk is evidently a one-valued function and that value is finite ; Sk is continuous ; and therefore the function Sk is uniform in the immediate vicinity of an ordinary point of the plane. 176 FUNCTIONS POSSESSING [99. For a point a, which is a branch-point of the function w, we know that the branches can be arranged in cyclical systems. Let w1,..., w^ be such a system. Then these branches interchange in cyclical order for a description of a small circuit round a ; and, if z — a = Z*, it is known (§ 93) that, in the vicinity of Z = 0, a branch w is a, uniform function of Z, say Therefore wk = Gk ) + Pk (Z) \£il in the vicinity of Z = 0 ; say w* = Ak + 2Bk>mZ-™ + 2 Ck>mZ™. m=l m=l Now the other branches of the function which are equal at a are derivable from any one of them by taking the successive values which that one acquires as the variable describes successive circuits round a. A circuit of w round a changes the argument of z — a, by 27r. and therefore gives Z reproduced but multiplied by a factor which is a primitive /xth root of unity, say by a factor a ; a second circuit will reproduce Z with a factor a2 ; and so on. Hence wf = Ak+2 Bk>m a— Z-™ +ZCk>m a- #» wrk = Ak+2 Bk>m a-™ Z~m + 2 Ck,m a™ Zm, m=l »»=! and therefore I* wrk = pAk + 2 Bkm - + ar + cr + . . . + cr't r=l m = 1 + 2 flto* Zm (1 + «m + a2"* + • • • + a""*-™). OT = 1 Now, since a is a primitive /*th root of unity, 1 +as + «2S+ ... + as('x-1) is zero for all integral values of s which are not integral multiples of p,, and it is yu, for those values of s which are integral values of jj, ; hence - £ B'k> i(z - a)"1 + B'k^(z - a)~2 + B'kt3 (z - a) . Hence the point z = a may be a singularity of 2 wrk but it is not a branch- r=l 99.] A FINITE NUMBER OF BRANCHES 177 point of the function ; and therefore in the immediate vicinity of z — a the *i quantity X wrk is a uniform function. r=l The point a is an essential singularity of this uniform function, if the order of the infinity of w at a be infinite : it is an accidental singularity, if that order be a finite integer. This result is evidently valid for all the cyclical systems at a, as well as for the individual branches which may happen to be one-valued at a. Hence (U. Sk, being the sum of sums of the form 2) wrk each of which is a uniform r=l function of z in the vicinity of a, is itself a uniform function of z in that vicinity. Also a is an essential singularity of Sk, if the order of the infinity at z = a for any one of the branches of w be infinite ; and it is an accidental singularity of Sk> if the order of the infinity at z = a for all the branches of w be finite. Lastly, it is an ordinary point of Sk, if there be no branch of w for which it is an infinity. Similarly for each of the branch-points. Again, let c be a regular singularity of any one (or more) of the branches of w ; then c is a regular singularity of every power of each of those branches, the singularities being simultaneously accidental or simultaneously essential. Hence c is a singularity of 8k : and therefore in the vicinity of c, $& is a uniform function, having c for an accidental singularity if it be so for each of the branches w affected by it, and having c for an essential singularity if it be so for any one of the branches w. It thus appears that in the part of the plane under consideration the function 8k is one-valued ; and it is continuous and finite, except at certain isolated points each of which is a singularity. It is therefore a uniform function in that part of the plane ; and the singularity of the function at any point is essential, if the order of the infinity for any one of the branches w at that point be infinite, but it is accidental, if the order of the infinity for all the branches w there be finite. And the number of these singularities is finite, being not greater than the combined number of the infinities of the function w, whether regular singularities or branch-points. Since the sums of the kth powers for all positive values of the integer k are uniform functions and since any integral symmetric function of the n values is a rational integral algebraical function of the sums of the powers, it follows that any integral symmetric function of the n values is a uniform function of z in the part of the plane under consideration ; and every infinity of a branch w leads to a singularity of the symmetric function, which is essential or accidental according as the orders of infinity of the various branches are not all finite or are all finite. F. 12 178 FUNCTIONS POSSESSING [99. Since w has n (and only n) values wlt... ,wn for each value of z, the equation which determines w is (W - Wj) (W-W2) ... (W- Wn) = 0. The coefficients of the various powers of w are symmetric functions of the branches wl , . . . , wn; and therefore they are uniform functions of z in the part of the plane under consideration. They possess a finite number of singularities, which are accidental or essential according to the character of the infinities of the branches at the same points. COROLLARY. If all the infinities of the branches in the finite part of the whole plane be of finite order, then the finite singularities of all the coefficients of the powers of w in the equation satisfied by w are all accidental ; and the coefficients themselves then take the form of a quotient of an integral uniform function (which may be either transcendental or algebraical, in the sense of § 47) by another function of a similar character. If z = oc be an essential singularity for at least one of the coefficients, through being an infinity of unlimited order for a branch of w, then one or both of the functions in the quotient-form of one at least of the coefficients must be transcendental. If z = oo be an accidental singularity or an ordinary point for all the coefficients, through being either an infinity of finite order or an ordinary point for the branches of w, then all the functions which occur in all the coefficients are rational, algebraical expressions. When the equation is multiplied throughout by the least common multiple of the denominators of the coefficients, it takes the form wnh0 (z) + wn~* A, (z) + . . . + w hn_, (z} + hn (z) = 0, where the functions h0(z), h^(z\ ..., hn(z) are rational, integral, algebraical functions of z, in the sense of § 47. A knowledge of the number of infinities of w gives an upper limit of the degree of the equation in z in the last form. Thus, let at be a regular singularity of the function ; and let Oi, fa, ji, ... be the orders of the infinities of the branches at at- ; then w^w-i ... wn(z — at )A', where \ denotes Oi + fti + % + ..., is finite (but not zero) for z = at. Let Ci be a branch-point, which is an infinity; and let p, branches w form a ft system for ct-, such that w(z — Cf)^ is finite (but not zero) at the point; then w:w2 ... Wp (z — Q) ' is finite (but not zero) at the point, and therefore also 99.] A FINITE NUMBER OF BRANCHES 179 is finite, where Qit (/>;, ^i, ... are numbers belonging to the various systems; or, if ei denote 0; + $f + tyi + . . . , then Wl...Wn(z- Ci)6i is finite for z = C;. Similarly for other symmetric functions of w. Hence, if «j, a2, ... be the regular singularities with numbers X1; X2, ... defined as above, and if c^ c2, ... be the branch -points, that are also infinities, with numbers e1; e2, ... defined as above, then the product (w-Wj) ...... (w-wn) n 0-af)A< n 0-Ci)e< i=l 1=1 is finite at all the points ai and at all the points c;. The points a and the points c are the only points in the finite part of the plane that can make the product infinite : hence it is finite everywhere in the finite part of the plane, and it is therefore an integral function of z. Lastly, let p be the number for z = oo corresponding to \i for af or to e^ for C;, so that for the coefficient of any power of w in (w — w^) ...(w — wn) the greatest difference in degree between the numerator and the denominator is p in favour of the excess of the former. Then the preceding product is of order which is therefore the order of the equation in z when it is expressed in a holomorphic form. 12—2 CHAPTER IX. PERIODS OF DEFINITE INTEGRALS, AND PERIODIC FUNCTIONS IN GENERAL. 100. INSTANCES have already occurred in which the value of a function of z is not dependent solely upon the value of z but depends also on the course of variation by which z obtains that value ; for example, integrals of uniform functions, and multiform functions. And it may be expected that, a fortiori, the value of an integral connected with a multiform function will depend upon the course of variation of the variable z. Now as integrals which arise in this way through multiform functions and, generally, integrals connected with differential equations are a fruitful source of new functions, it is desirable that the effects on the value of an integral caused by variations of a £-path be assigned so that, within the limits of algebraic possibility, the expression of the integral may be made completely determinate. There are two methods which, more easily than others, secure this result ; one of them is substantially due to Cauchy, the other to Riemann. The consideration of Riemann's method, both for multiform functions and for integrals of such functions, will be undertaken later, in Chapters XV., XVI. Cauchy's method has already been used in preceding sections relating to uniform functions, and it can be extended to multiform functions. Its characteristic feature is the isolation of critical points, whether regular singularities or branch-points, by means of small curves each containing one and only one critical point. Over the rest of the plane the variable z ranges freely and, under certain conditions, any path of variation of z from one point to another can, as will be proved immediately, be deformed without causing any change in the value of the integral, provided that the path does not meet any of the small curves in the course of the deformation. Further, from a knowledge of the relation of any point thus isolated to the function, it is possible to calculate the change caused by a deformation of the £-path over such a point; and thus, for defined deformations, the value of the integral can be assigned precisely. 100.] INTEGRAL OF A BRANCH 181 The properties proved in Chapter II. are useful in the consideration of the integrals of uniform functions ; it is now necessary to establish the propositions which give the effects of deformation of path on the integrals of multiform function. The most important of these propositions is the following : — fb If w be a multiform function, the value of I wdz, taken between two J a ordinary points, is unaltered for a deformation of the path, provided that the initial branch of w be the same and that no branch-point or infinity be crossed in the deformation. Consider two paths acb, adb, (fig. 16, p. 152), satisfying the conditions specified in the proposition. Then in the area between them the branch w has no infinity and no point of discontinuity ; and there is no branch-point in that area. Hence, by § 90, Corollary VI., the branch w is a uniform monogenic function for that area; it is continuous and finite everywhere within it and, by the same Corollary, we may treat w as a uniform, mono genic, finite and continuous function. Hence, by § 17, we have rb ra (c) I wdz + (d) wdz = 0, J a J b the first integral being taken along acb and the second along bda; and therefore rb ra rb (c) wdz = — (d}\ wdz = (d) \ wdz, Jo, J b J a shewing that the values of the integral along the two paths are the same under the specified conditions. It is evident that, if some critical point be crossed in the deformation, the branch w cannot be declared uniform and finite in the area and the theorem of § 17 cannot then be applied. COROLLARY I. The integral round a closed curve containing no critical point is zero. COROLLARY II. A curve round a branch-point, containing no other critical point of the function, can be deformed into a loop without altering the value of fwdz ; for the deformation satisfies the condition of the proposition. Hence, when the value of the integral for the loop is known, the value of the integral is known for the curve. COROLLARY III. From the proposition it is possible to infer conditions, under which the integral fwdz round the whole of any curve remains unchanged, when the whole curve is deformed, without leaving an infinitesimal arc common as in Corollary II. 182 INTEGRATION [100. Let GDC', ABA' be the curves: join two consecutive points A A' to two consecutive points (7(7. Then if the area CABA'C'DG enclose no critical point of the function w, the value of jwdz along CDC' is by the proposition the same as its value along CABA'C'. The latter is made up of the value along CA, the value along ABA', and the value along AC', say rA r rC' I wdz + I wdz + w'dz, v. Jc JB JA «§. where w' is the changed value of w consequent on the description of a simple curve reducible to B (§ 90, Cor. II.). Now since w is finite everywhere, the difference between the values of w at A and at A' consequent on the description of ABA is finite : hence as A A is infinitesimal the value of jwdz necessary to complete the value for the whole curve B is infinitesimal and therefore the complete value can be taken as the foregoing integral wdz. Similarly for the complete value J B along the curve D : and therefore the difference of the integrals round B and round D is rA rC' I wdz + I w'dz, J C J A' rA say (w — w') dz. J c In general this integral is not zero, so that the values of the integral round B and round D are not equal to one another : and therefore the curve D cannot be deformed into the curve B without affecting the value of jwdz round the whole curve, even when the deformation does not cause the curve to pass over a critical point of the function. But in special cases it may vanish. The most important and, as a matter of fact, the one of most frequent occurrence is that in which the description of the curve B restores at A' the initial value of w at A. It easily follows, by the use of § 90, Cor. II., that the description of D (as suming that the area between B and D includes no critical point) restores at C' the initial value of w at (7. In such a case, w = w' for corresponding points on AC and A'C', and the integral, which expresses the difference, is zero: the value of the integral for the curve B is then the same as that for D. Hence we have the proposition : — If a curve be such that the description of it by the independent variable restores the initial value of a multiform function w, then the value of jwdz taken round the curve is unaltered when the curve is deformed into any other curve, provided that no branch-point or point of discontinuity of w is crossed in the course of deformation. 100.] OF MULTIFORM FUNCTIONS 183 This is the generalisation of the proposition of § 19 which has thus far been used only for uniform functions. Note. Two particular cases, which are very simple, may be mentioned here : special examples will be given immediately. The first is that in which the curve B, and therefore also D, encloses no branch-point or infinity; the initial value of w is restored after a description of either curve, and it is easy to see (by reducing B to a point, as may be done) that the value of the integral is zero. The second is that in which the curve encloses more than one branch point, the enclosed branch-points being such that a circuit of all the loops, into which (by Corollary V., § 90) the curve can be deformed, restores the initial branch of w. This case is of especial importance when w is two-valued : the curves then enclose an even number of branch-points. 101. It is important to know the value of the integral of a multiform function round a small curve enclosing a branch-point. Let c be a point at which TO branches of an algebraical function are equal and interchange in a single cycle ; and let c, if an infinity, be of only finite order, say k/m. Then in the vicinity of c, any of the branches w can be expressed in the form 00 .« w= 2 gs(z-c)m, o — If o — — K where k is a finite integer. The value of jwdz taken round a small curve enclosing c is the sum of the integrals the value of which, taken once round the curve and beginning at a point zly is TO + S where a is a primitive mth root of unity, provided TO + s is not zero. If then s + m be positive, the value is zero in the limit when the curve is infini tesimal : if TO + s be negative, the value is oo in the limit. But, if m + s be zero, the value is Z7rigs. Hence we have the proposition: If, in the vicinity of a branch-point c, where m branches w are equal to one another and interchange cyclically, the expression of one of the branches be 184 MULTIPLICITY OF VALUE [101. then jwdz, taken once round a small curve enclosing c, is zero, if k<m; is infinite, if k> m ; and is ^irig^ , if k = m. It is easy to see that, if the integral be taken m times round the small curve enclosing c, then the value of the integral is 2m7rigm when k is greater than in, so that the integral vanishes unless there be a term involving (z — c)"1 in the expansion of a branch w in the vicinity of the point. The reason that the integral, which can furnish an infinite value for a single circuit, ceases to _* do so for m circuits, is that the quantity (^ — c) m, which becomes indefi nitely great in the limit, is multiplied for a single circuit by a*— 1, for a second circuit by a2A — aA, and so on, and for the mth circuit by awA — a(w~1)A, the sum of all of which coefficients is zero. Ex. The integral \{(z - a) (z - b) ... (z -f)}~* dz taken round an indefinitely small curve enclosing a is zero, provided no one of the quantities b, ... ,/ is equal to a. 102. Some illustrations have already been given in Chapter II., but they relate solely to definite, not to indefinite, integrals of uniform functions. The whole theory will not be considered at this stage ; we shall merely give some additional illustrations, which will shew how the method can be applied to indefinite integrals of uniform functions and to integrals of multiform functions, and which will also form a simple and convenient introduction to the theory of periodic functions of a single variable. We shall first consider indefinite integrals of uniform functions. f dz Ex. 1. Consider the integral I — , and denote* it by/ (z}. The function to be integrated is uniform, and it has an accidental singularity of the first order at the origin, which is its only singularity. The value of \z~l dz taken positively along a small curve round the origin, say round a circle with the origin as centre, is 2n-i • but the value of the integral is zero when taken along any closed curve which does not include the origin. Taking z = l as the lower limit of the integral, and any point z as the upper limit, we consider the possible paths from 1 to z. Any path from 1 to z can be deformed, without crossing the origin, into a path which circumscribes the origin positively some number of times, say m^, and negatively some number of times, say »i2, all in any order, and then leads in a straight line from 1 to z. For this path the value of the integral is equal to I — , J 1 z that is, to 2mni+ I — , Ji z where m is an integer, and in the last integral the variation of z is along a straight line from 1 to z. Let the last integral be denoted by u ; then * See Chrystal, ii, pp. 266 — 272, for the elementary properties of the function and its inverse, when the variable is complex. 102.] OF INTEGRALS 185 and therefore, inverting the function and denoting/"1 by <j>, we have Hence the general integral is a function of z with an infinite number of values ; and z is a periodic function of the integral, the period being 2n-z. Ex. 2. Consider the function / - - ^ > and again denote it by / (z). The one- valued function to be integrated has two accidental singularities + i, each of the first order. The value of the integral taken positively along a small curve round i is TT, and along a small curve round — i is — n. We take the origin 0 as the lower limit and any point z as the upper limit. Any path from 0 to z can be deformed, without crossing either of the singularities and therefore without changing the value of the integral, into (i) any numbers of positive (ml5 w?2) an(* of negative (nz/, m2') circuits round i and round -i, and (ii) a straight line from 0 to z. Then we have - TJ-) +WIJJ ( - IT) + m.2' {_(-„•)}+ /* . J o , z = nir+ where ?i is an integer and the integral on the right-hand side is taken along a straight line from 0 to z. Inverting the function and denoting/"1 by tp, we have The integral, as before, is a function of z with an infinite number of values ; and z is a periodic function of the integral, the period being TT. 103. Before passing to the integrals of multiform functions, it is con venient to consider the method in which Hermite* discusses the multiplicity in value of a definite integral of a uniform function. Taking a simple case, let <£> (X) = \ J Q 1 + Z and introduce a new variable t such that Z—zt\ then zdt When the path of t is assigned, the integral is definite, finite and unique in value for all points of the plane except for those for which 1 + zt = 0 ; and, according to the path of variation of t from 0 to 1, there will be a 0-curve which is a curve of discontinuity for the subject of integration. Suppose the path of t to be the straight line from 0 to 1 ; then the curve of discontinuity * Crelle, t. xci, (1881), pp. 62—77; Cours a la Faculte des Sciences, 46me 6d. (1891), pp. 76—79, 154—164, and elsewhere. 186 HERMITE'S [103. is the axis of x between — 1 and — oo . In this curve let any point - £ be taken where £ > 1 ; and consider a point z1 — -^ + ie and a point z2 = — £ — ie, respectively on the positive and the negative sides of the axis of x, both being ultimately taken as infinitesimally near the point — £. Then dt= ( Let e become infinitesimal ; then, when t is infinite, we have tan for e is positive ; and, when t is unity, we have tan"1 ----- = — |TT, for £ is > 1. Hence <£ (^) — <£ (^2) The part of the axis of x from - 1 to - oo is therefore a line of discon tinuity in value of <j> (z), such that there is a sudden change in passing from one edge of it to the other. If the plane be cut along this line so that it cannot be crossed by the variable which may not pass out of the plane, then the integral is everywhere finite and uniform in the modified surface. If the plane be not cut along the line, it is evident that a single passage across the line from one edge to the other makes a difference of 2?ri in the value, and consequently any number of passages across will give rise to the multiplicity in value of the integral. Such a line is called a section* by Hermite, after Riemann who, in a different manner, introduces these lines of singularity into his method of representing the variable on surfaces "f*. When we take the general integral of a uniform function of Z and make the substitution Z = zt, the integral that arises for consideration is of the form We shall suppose that the path of variation of t is the axis of real quantities : and the subject of integration will be taken to be a general function of t and z, without special regard to its derivation from a uniform function of Z. * Coupure; see Crelle, t. xci, p. 62. t See Chapter XV. 103.] SECTIONS 187 It is easy, after the special example, to see that ^ is a continuous function of z in any space that does not include a ^-point which, for values of t included within the range of integration, would satisfy the equation. G (t, z) = 0. But in the vicinity of a ^-point, say £, corresponding to the value t = 6 in the range of integration, there will be discontinuity in the subject of integration and also, as will now be proved, in the value of the integral. Let Z be the point £ and draw the curve through Z corresponding to t = real constant ; let Nt be a point on the positive side and N2 a point on the negative side of this curve positively described, both points being on the normal at Z ; and let supposed small. Then for N! we have X-L = g — e sin y, yl = ^-\-e cos y , Fig. 24. so that z1 = £+16' (cosy + isiny), where ty is the inclination of the tangent to the axis of real quantities. But, if da- be an arc of the curve at Z, da , • • i \ d% • dt] d£ for variations along the tangent at Z, that is, i da- . . 3 -j-- (cos y + i sin y ) = — - Thus, since -j- may be taken as finite on the supposition that Z is an ordinary point of the curve, we have where e = e' -y- , P = - Similarly z.2 = £ + ie -^r. Hence <1> (^) = I --i-i— *£ ^ w/ n_^J_w/ m _ 1*. 188 HERMITE'S [103. with a similar expression for <& (z2) ; and therefore F(t, 0 j- [G (t, ®}^-G (t, §) ' The subject of integration is infinitesimal, except in the immediate vicinity of t = 6 ; and there powers of small quantities other than those retained being negligible. Let the limiting values of t, that need be retained, be denoted by d + v and d — p', then, after reduction, we have edt F(e, in the limit when e is made infinitesimal. Hence a line of discontinuity of the subject of integration is a section for the integral ; and the preceding expression is the magnitude, by numerical multiples of which the values of the integral differ*. Ex. 1. Consider the integral dZ / zdt h We have S ^ *' =^ = ^g = ^. so that TT is the period for the above integral. Ex. 2. Shew that the sections for the integral ta sin z , 2 ' * The memoir and the Cmirs d' 'Analyse of Hermite should be consulted for further develop ments; and, in reference to the integral treated above, Jordan, Cours d' Analyse, t. iii, pp. 610 — 614, may be consulted with advantage. See also, generally, for functions defined by definite integrals, Goursat, Acta Math., t. ii, (1883), pp. 1—70, and ib., t. v, (1884), pp. 97— 120; and Pochhammer, Math. Arm., t. xxxv, (1890), pp. 470—494, 495—526. Goursat also discusses double integrals. 103.] SECTIONS 189 where a is positive and less than 1, are the straight lines x = (2k + l) TT, where k assumes all integral values ; and that the period of the integral at any section at a distance 77 from the axis of real quantities is 2?r cosh (arj). (Hermite.) Ex. 3. Shew that the integral o where the real parts of /3 and y — /3 are positive, has the part of the axis of real quantities between 1 and +00 for a section. Shew also that the integral i rht }— (z P~I (~i - vy~'3~1n— }~a d J 0 where the real parts of /3 and 1 - a are positive, has the part of the axis of real quantities between 0 and 1 for a section : but that, in order to render <£ (z) a uniform function of z, it is necessary to prevent the variable from crossing, not merely the section, but also the part of the axis of real quantities between 1 and + <x> . (Goursat.) (The latter line is called a section of the second kind.) Ex. 4. Discuss generally the effect of changing the path of t on a section of the integral ; and, in particular, obtain the section for I — „ when, after the substitution jo 1 + « Z=zt, the path of t is made a semi-circle on the line joining 0 and 1 as diameter. Note. It is manifestly impossible to discuss all the important bearings of theorems and principles, which arise from time to time in our subject ; we can do no more than mention the subject of those definite integrals involving complex variables, which first occur as solutions of the better-known linear differential equations of the second order. Thus for the definite integral connected with the hypergeometric series, memoirs by Jacobi* and Goursat t should be consulted ; for the definite integral connected with Bessel's functions, memoirs by HankelJ and Weber § should be consulted ; and Heine's J/andbuch der Kugelfunctionen for the definite integrals connected with Legendre's functions. 104. We shall now consider integrals of multiform functions. Ex. 1. To find the integral of a multiform function round one loop ; and round a number of loops. Let the function be i w={(z-al}(z-a.z}...(z- an)}»» , where m may be a negative or positive integer, and the quantities a are unequal to one another ; and let the loop be from the origin round the point ax. Then, if / be the value of the integral with an assigned initial branch w, we have /a, f CO wdz-\- I wdz + I awdz, 0 J c J a. where a is e m and the middle integral is taken round the circle at a^ of infinitesimal radius. * Crelle, t. Ivi, (1859), pp. 149 — 165 ; the memoir was not published until after his death, t Sur Vequation differentielle lineaire qui admet pour integrate la serie hypergrometrique, (These, Gauthier-Villars, Paris, 1881). I Math. Ann., t. i, (1869), pp. 467—501. § Math. Ann., t. xxxvii, (1890), pp. 404—416. 190 EXAMPLES [104. But, since the limit of (z-ajw when z = a1 is zero, the middle integral vanishes by § 101 ; and therefore /"«i «, = (! -a) I web, Jo where the integral may, if convenient, be considered as taken along the straight line from 0 to al . (2) (3) Fig. 25. Next, consider a circuit for an integral of w which (fig. 25) encloses two branch-points, say «! and «2, but no others ; the circuit in (1) can be deformed into that in (2) or into that in (3) as well as into other forms. Hence the integral round all the three circuits must be the same. Beginning with the same branch as in the first case, we have (1 /«! wdz, o as the integral after the first loop in (2). And the branch with which the second loop begins is aw, so that the integral described as in the second loop is /«2 awdz; 0 and therefore, for the circuit as in (2), the integral is Cat [ay 1= (1 - a) I wdz + a (1 - a) / wdz. Jo Jo Proceeding similarly with the integral for the circuit in (3), we find that its expression is /a2 /"<*! wdz + a (I -a) I wdz, 0 J 0 and these two values must be equal. But the integrals denoted by the same symbols are not the same in the two cases ; the function I * wdz is different in the second value of J from that in the first, for the deforma- Jo tion of path necessary to change from the one to the other passes over the branch-point az. In fact, the equality of the two values of / really determines the value of the integral for the loop Oal in (3). And, in general, equations thus obtained by varied deformations do not give relations among loop-integrals but define the values of those loop-integrals for the deformed paths. We therefore take that deformation of the circuit into loops which gives the simplest path. Usually the path is changed into a group of loops round the branch-points as they occur, taken in order in a trigonometrically positive direction. The value of the integral round a circuit, equivalent to any number of loops, is obvious. Ex. 2. To find the value of $wdz, taken round a simple curve which includes all the branch-points of w and all the infinities. 104.] OF PERIODICITY OF INTEGRALS 191 If z = oo be a branch-point or an infinity, then all the branch-points and all the infinities of w lie on what is usually regarded as the exterior of the curve, or the curve may in one sense be said to exclude all these points. The integral round the curve is then the integral of a function round a curve, such that over the area included by it the function is uniform, finite and continuous ; hence the integral is zero. If 0 = 00 be neither a branch-point nor an infinity, the curve can be deformed until it is a circle, centre the origin and of very great radius. If then the limit of zw, when \z is infinitely great, be zero, the value of the integral again is zero, by II., § 24. Another method of considering the integral, is to use Neumann's sphere for the representation of the variable. Any simple closed curve divides the area of the sphere into two parts ; when the curve is defined as above, one of those parts is such that the function is uniform, finite and continuous throughout and therefore its integral round the curve, regarded as the boundary of that part, is zero. (See Corollary III., § 90.) Ex. 3. To find the general value of J(l-22)~*cfe. The function to be integrated is two-valued: the two values interchange round each of the branch-points ±1, which are the only branch-points of the function. Let / be the value of the integral for a loop from the origin round +1, beginning with the branch which has the value +1 at the origin ; and let /' be the corresponding value for the loop from the origin round - 1, beginning with the same branch. Then, by Ex. 1, /= 2 P (1 - z*T*dz, /' = 2 f"1 (1 - z2)"* dz = -/, the last equality being easily obtained by changing variables. Now consider the integral when taken round a circle, centre the origin and of indefinitely great radius R ; then by § 24, II., if the limit of zw for z= QO be k, the value of \wdz round this circle is 2iri&. In the present case w = (l- 22)~^ so that the limit of zw is + ^ ; hence J(l-22r^2 = 27T, the integral being taken round the circle. But since a description of the circle restores the initial value, it can be deformed into the two loops from 0 O' to A and from 0 to A'. The value round the first is /; and ^ r > ^ the branch with which the second begins to be described has the value — 1 at the origin, so that the consequent value round *1S- ^"- the second is — /' ; hence 7-/' = 2»r* and therefore verifying the ordinary result that when the integral is taken along a straight line. To find the general value of u for any path of variation between 0 and z, we proceed as follows. Let Q be any circuit which restores the initial branch of (l-z2)~^. Then by § 100, Corollary II., Q may be composed of (i) a set of double circuits round + 1, say m', (ii) a set of double circuits round - 1, say m", and (iii) a set of circuits round + 1 and - 1 ; * It is interesting to obtain this equation when O' is taken as the initial point, instead of 0. 192 EXAMPLES OF PERIODICITY [104. and these may come in any order and each may be described in either direction. Now for a double circuit positively described, the value of the integral for the first description is / and for the second description, which begins with the branch —(1 — z2)~^, it is — /; hence for the double circuit it is zero when positively described, and therefore it is zero also when negatively described. Hence each of the TO' double circuits yields zero as its nett contribu tion to the integral. Similarly, each of the m" double circuits round - 1 yields zero as its nett contribution to the integral. For a circuit round + 1 and - 1 described positively, the value of the integral has just been proved to be /-/', and therefore when described negatively it is /'-/. Hence if there be n^ positive descriptions and n2 negative descriptions, the nett contribution of all these circuits to the value of the integral is (n± — n^) (I - 1'), that is, 2nir where n is an inteer. Hence the complete value for the circuit Q i Now any path from 0 to z can be resolved into a circuit Q, which restores the initial branch of (1 — 22)~ , chosen to have the value + 1 at the origin, and either (i) a straight line Oz ; or (ii) the path OACz, viz., a loop round + 1 and the line Oz ; or (iii) the path OA'Cz, viz., a loop round - 1 and the line Oz. Let u denote the value for the line Oz, so that u= f* (!-#)-* dk. J o Hence, for case (i), the general value of the integral is 2W7T + U. For the path OA Cz, the value is 7 for the loop OAC, and is ( — u) for the line Cz, the negative sign occurring because, after the loop, the branch of the function for integra tion along the line is —(1 — 22)~5 ; this value is I—u, that is, it is TT — U. Hence, for case (ii), the value of the integral is — U. For the path OA'Cz, the value is similarly found to be - TT - u ; and therefore, for case (iii), the value of the integral is 2?wr — ir-u. If /(z) denote the general value of the integral, we have either Or /(Z) = (2TO+1)7T-W, where n and m are any integers, so that/ (z) is a function with two infinite series of values. Lastly, if z = $($) be the inverse oif(z} = 6, then the relation between u and z given by can be represented in the form and 104.] OF INTEGRALS 193 both equations being necessary for the full representation. Evidently z is a simply -periodic function of u, the period being 2?r ; and from the definition it is easily seen to be an odd function. Let y = (\ -z2)—x (u\ so that y is an even function of u ; from the consideration of the various paths from 0 to 2, it is easy to prove that Ex. 4. To find the general value of f{(l-j*)(l-IM)}~*dk It will be convenient (following Jordan *) to regard this integral as a special case of Z= \{(z -a)(z- b) (z -c}(z- d)}~* dz = \wdz. The two-valued function to be integrated has a, 6, c, d (but not oo ) as the complete system of branch-points ; and the two values interchange at each of them. We proceed as in the last example, omitting mere re-statements of reasons there given that are applicable also in the present example. Any circuit Q, which restores an initial branch of w, can be made up of (i) sets of double circuits round each of the branch-points, and (ii) sets of circuits round any two of the branch -points. The value of \wdz for a loop from the origin to a branch-point k (where k = a,b, c, or d) is 2 I wdz ; J o and this may be denoted by K, where K=A, B, C, or D. The value of the integral for a double circuit round a branch-point is zero. Hence the amount contributed to the value of the integral by all the sets in (i) as this part of Q is zero. The value of the integral for a circuit round a and b taken positively is A - B ; for one round b and c is B- C ; for one round c and d is C-D; for one round a and c is A - C, which is the sum of A - B and B-C; and similarly for circuits round a and d and round b and d. There are therefore three distinct values, say A-B, B-C, C-D, the values for circuits round a and b, b and c, c and d respectively ; the values for circuits round any other pair can be expressed linearly in terms of these values. Suppose then that the part of Q represented by (ii), when thus resolved, is the nett equivalent of the description of m' circuits round a and b, of n' circuits round b and c, and of I' circuits round c and d. Then the value of the integral contributed by this part of Q is • which is therefore the whole value of the integral for Q. But the values of A, £, C, D are not independent f. Let a circle with centre the origin and very great radius be drawn ; then since the limit of zw for |s| = oo is zero and since 2= cc is not a branch-point, the value of \wdz round this circle is zero (Ex. 2). The circle can be deformed into four loops round a, b, c, d respectively in order ; and therefore the value of the integral is A - B + C- D, that is, Hence the value of the integral for the circuit fl is where m and n denote m' - 1' and n' - 1' respectively. * Cours d' Analyse, t. ii, p. 343. t For a purely analytical proof of the following relation, see Greenhill's Elliptic Functions Chapter II. F- 13 194 PERIODICITY [104. Now any path from the origin to z can be resolved into Q, together with either (i) a straight line from 0 to z, or (ii) a loop round a and then a straight line to z. It might appear that another resolution would be given by a combination of Q with, say, a loop round b and then a straight line to z ; but it is resoluble into the second of the above combinations. For at C, after the description of the loop B , introduce a double description of the loop A, which adds nothing to the value of the integral and does not in the end affect the branch of w at C ; then the new path can be regarded as made up of (a) the circuit constituted by the loop round b and the first loop round a, (/3) the second loop round a, which begins with the initial branch of w, followed by a straight path to z. Of these (a) can be absorbed into G, and (/3) is the same as (ii) ; hence the path is not essentially new. Similarly for the other points. Let u denote the value of the integral with a straight path from 0 to z; then the whole value of the integral for the combination of Q with (i) is of the form For the combination of O with (ii), the value of the integral for the part (ii) of the path is J, for the loop round a, +(-«), for the straight path which, owing to the description of the loop round a, begins with - w ; hence the whole value of the integral is of the form Hence, if / (z) denote the general value of the integral, it has two systems of values, each containing a doubly -infinite number of terms; and, if z = <j>(u) denote the inverse of u = f (z\ we have = 0 {m (A-B} + n(B-C)+A - u}, where m and n are any integers. Evidently z is a doubly-periodic function of u, with periods A-B and B-C. Ex. 5. The case of the foregoing integral which most frequently occurs is the elliptic integral in the form used by Legendre and Jacobi, viz. : u = J{(1 - z2) (1 - kW)}-*dz = \wdz, where k is real. The branch-points of the function to be integrated are 1, -1, ^ and -L and the values of the integral for the corresponding loops from the origin are A/ A 2 I wdz, J o r-i ri 2 I wdz— -2 I wdz, Jo /• I wdz, '' and Now the values for the loops are connected by the equation * The value for a loop round b and then a straight line to z, just considered, is B - u = -(A-B) + being the value in the text with m changed to m - 1. 104.] OF ELLIPTIC INTEGRALS 195 and so it will be convenient that, as all the points lie on the axis of real variables, we arrange the order of the loops so that this relation is identically satisfied. Otherwise, the relation will, after Ex. 1, be a definition of the paths of integration chosen for the loops. Among the methods of arrangement, which secure the identical satisfaction of the Fig. 28. relation, the two in the figure* are the simplest, the curved lines being taken straight in the limit ; for, by the first arrangement when k < 1, we have and, by the second when £ > 1, we have both of which are identically satisfied. We may therefore take either of them ; let the former be adopted. The periods are A-B, B-C, (and C-D, which is equal to B-A\ and any linear combination of these is a period: we shall take A - B, and B-D. The latter, B-D, is equal to n r-i 2 / wdz -2 I wdz, Jo Jo which, being denoted by 4/f, gives 4J5T=4 / JO{(1-22)(1_£222)}4 as one period. The former, A-B, is equal to 2 I wdz -2 I wdz, Jo Jo i / wdz; /k 1|(1- which is 2 this, being denoted by 2iK', gives dz dz' where £'2 + £2=l and the relation between the variables of the integrals is i Hence the periods of the integral are 4K and ZiK'. Moreover, A is 2 I" wdz, which i i J» 2 / wdz + 2 I wdz = Jo J i Hence the general value of f* {(I - z*) (I - * Jordan, Cours d' 'Analyse, t. ii, p. 356. 13—2 196 PERIODICITY [104. or that is, 2K-u + 4mK+2niK', where u is the integral taken from 0 to z along an assigned path, often taken to be a straight line ; so that there are two systems of values for the integral, each containing a doubly -infinite number of terms. If z be denoted by $ (u) — evidently, from the integral definition, an odd function of u — , then so that z is a doubly-periodic function of u, the periods being 4A and 2iK'. Now consider the function ^ = (1 -zrf. A 2-path round T does not affect ^ by way of change, provided the curve does not include the point 1 ; hence, if zt = x (u), we have But a z- path round the point 1 does change % into —z1; so that X («)--* («+**} Hence x (u\ which is an even function, has two periods, viz., 4AT and 2A' + 2i'A", whence x(u) = x(u + 4mK+ 2nK+ 2niK'). Similarly, taking z2 = (l -Fs2)* = -f (u), it is easy to see that so that ^ (u), which is an even function, has two periods, viz., 2 A' and 4iK' ; whence = u The functions <£ (u), x (u\ ^ (M) are of course sn w> cn '""> dn M respectively. Ex. 6. To find the general value of the integral The function to be integrated has e^ e2, e3, and co for its branch-points; and for paths round each of them the two branches interchange. A circuit G which restores the initial branch of the function to be integrated can be resolved into : — (i) Sets of double circuits round each of the branch-points alone : as before, the value of the integral for each of these double circuits is zero. (ii) Sets of circuits, each enclosing two of the branch-points : it is convenient to retain circuits including oo and en oo and e.2, oc and e3, the other three combinations being reducible to these. The values of the integral for these three retained are respectively E! = 2 f (4 (z - ej (z - e2} (z - e^dz = 2«1 , J «i Ez=2 I {4(2-e1)(2-e2)(s-e3)}~ick=2a>2, J 62 3 J ea * The choice of o> for the upper limit is made on a ground which will subsequently be considered, viz., that, when the integral is zero, z is infinite. 104.] OF ELLIPTIC INTEGRALS 197 and therefore the value of the integral for the circuit O is of the form But E^ K2, E3 are not linearly independent. The integral of the function round any curve in the finite part of the plane, which does not include el5 e<2 or e3 within its boundary, is zero, by Ex. 2; and this curve can be deformed to the shape in the figure, until it becomes infinitely large, without changing the value of the integral. Since the limit of zw for \z\ = 00 is zero, the value of the integral from oo ' to oo is zero, by § 24, II. ; and if the description begin with a branch w, the branch at oo is -w. The rest of the integral consists of the sum of the values Fig. 29. round the loops, which is because a path round a loop changes the branch of w and the last branch after describing the loop round e3 is +w at GO', the proper value (§ 90, in). Hence, as the whole integral is zero, we have or say E2 = Thus the value of the integral for any circuit Q, which restores the initial branch of w, can be expressed in any of the equivalent forms mE^ n E3, m'E^n'E^ m"E2 + ri'Ez, where the m's and ris, are integers. Now any path from co to z can be resolved into a circuit fl, which restores at oo the initial branch of w, combined with either (i) a straight path from oo to 2, or (ii) a loop between oo and e1} together with a straight path from oo to z. ' (The apparently distinct alternatives, of a loop between oo and e2 together with a straight : path from oo to z and of a similar path round ea, are inclusible in the second alternative above ; the reasons are similar to those in Ex. 5.) fx If u denote j ^ {^(z-ej (z-e2) (z-e3}}~*dz when the integral is taken in a straight , line, then the value of the integral for part (i) of a path is u; and the value of the 1 integral for part (ii) of a path is El - u, the initial branch in each case for these parts being . the initial branch of w for the whole path. Hence the most general value of the integral for any path is + 2no>3 + u, or the two being evidently included in the form 2mo>1 + 2n(,)3±u. If, then, we denote by z = ft>(u) the relation which is inverse to we In the same way as in the preceding example, it follows that where ^ («) is - {4 (z - e^ (z - e2) (z - e3)}*. 198 SIMPLE PERIODICITY [104 The foregoing simple examples are sufficient illustrations of the multi plicity of value of an integral of a uniform function or of a multiform function, when branch-points or discontinuities occur in the part of the plane in which the path of integration lies. They also shew one of the modes in which singly-periodic and doubly-periodic functions arise, the periodicity consisting in the addition of arithmetical multiples of constant quantities to the argument. And it is to be noted that, as only a single value of z is used in the integration, so only a single value of z occurs in the inversion ; that is, the functions just obtained are uniform functions of their variables. To the properties of such periodic functions we shall return in the succeeding chapters. 105. We proceed to the theory of uniform periodic functions, some special examples of which have just been considered ; and limitation will be made here to periodicity of the linear additive type, which is only a very special form of periodicity. A function f(z) is said to be periodic when there is a quantity &> such that the equation /(* + »)=/(*) is an identity for all values of z. Then/0 + nw) =f(z), where n is any integer positive or negative; and it is assumed that &> is the smallest quantity for which the equation holds, that is, that no submultiple of &> will satisfy the equation. The quantity u> is called a period of the function. A function is said to be simply-periodic when there is only a single period : to be doubly-periodic when there are two periods ; and so on, the periodicity being for the present limited to additive modification of the argument. It is convenient to have a graphical representation of the periodicity of a function. (i) For simply-periodic functions, we take a series of points 0, A1} A2,..., A-i, ^4_2,... representing 0, w, 2o>, ... , — <», — 2&>, . . . ; and through these points we draw a series of parallel lines, dividing the plane into bands. Let P be any point z in the band between the lines through 0 and through A^\ through P draw a line parallel to OAl and measure each equal to OA^ then all the points / P1} P2, ... , P_i, P-2, ... are represented by z + nco for positive and negative integral values of n. But/ (2 + &»)=/(•*)] and therefore the value of the function at a point Pn in any of the bands is 105.] DOUBLE PERIODICITY 199 the same as the value at P. Moreover to a point in any of the bands there corresponds a point in any other of the bands ; and therefore, owing to the periodic resumption of the value at the points corresponding to each point P, it is sufficient to consider the variation of the function for points within one band, say the band between the lines through 0 and through AI. A point P within the band is sometimes called irreducible, the corresponding points P in the other bands reducible. If it were convenient, the boundary lines of the bands could be taken through points other than Al} A2, ... ; for example, through points (m + |) &> for positive and negative integral values of ra. Moreover, they need not be straight lines. The essential feature of the graphic representation is the division of the plane into bands. (ii) For doubly-periodic functions a similar method is adopted. Let &> and co' be the two periods of such a function /(#), so that /<«.+»)»/(*)-/(•+ <0; then f(z + nw + n'w) =f(z), where n and n' are any integers positive or negative. For graphic purposes, we take points 0, A-L, A2, ..., A^i, A_2, ... representing 0, ft), 2&), . . . , — to, — 2(w, . . . ; and we take another series 0, B1} B2, . . . , B_1} B_2, . . . representing 0, &)', 2&/, . . . , — ft/, — 2ft/, . . . ; through the points A we draw lines parallel to the line of points B, and through the points B we draw lines parallel to the line of points A. The intersection of the lines through An and Bn> is evidently the point n&> + w'&>', that is, the angular points of the parallelograms into which the plane is divided represent the points nco + n'w for the values of n and n'. Let P be any point z in the parallelogram OAfi-JS^ ; on lines through P, parallel to the sides of the parallelogram, take points Q1} Q2, ... , Q_lt Q_2, ... such that PQl = QiQ2= ... = ft) and points Rlt R2, ... , R_lt R_2, ... such that PRl = R^ — . . . — to' ; and through these new points draw lines parallel to the sides of the parallelogram. Then the variables of the points in which these lines intersect are all represented by z + mw + mV for positive and nega tive integral values of m and m' ; and the point represented by z + m^ + m'a)' is situated in the parallelogram, the angular points of which are mw 4 mot', (m + 1) &) + mw, mco + (mf + 1) ft)', and (m -f 1) &) + (m + 1) ft/, exactly as P is situated in OA^C^. But / (z + m^ + Wj V) = / (z\ Fig. 31. 200 RATIO OF THE PERIODS [105. and therefore the value of the function at such a point is the same as the value at P. Since the parallelograms are all equal and similarly situated. to any point in any of them there corresponds a point in OA^G^B^; and the value of the function at the two points is the same. Hence it is sufficient to consider the variation of the function for points within one parallelogram, say, that which has 0, &>, o) + «', &>' for its angular points. A point P within this parallelogram is sometimes called irreducible, the corresponding points within the other parallelograms reducible to P ; the whole aggregate of the points thus reducible to any one are called homologous points. And the parallelogram to which the reduction is made is called the parallelogram of periods. As in the case of simply-periodic functions, it may prove convenient to choose the position of the fundamental parallelogram so that the origin is not on its boundary ; thus it might be the parallelogram the middle points of whose sides are + £&>, + ^co'. 106. In the preceding representation it has been assumed that the line of points A is different in direction from the line of points B. If &> = u + iv and to' = u'+iv', this assumption implies that v'/u' is unequal to v/u, and therefore that the real part of a>'/ia> does not vanish. The justification of this assumption is established by the proposition, due to Jacobi * : — The ratio of the periods of a uniform doubly -periodic function cannot be real. Let/ (2) be a function, having CD and CD' as its periods. If the ratio w'/to be real, it must be either commensurable or incommensurable. If it be commensurable, let it be equal to n'/n, where n and n' are integers, neither of which is unity owing to the definition of the periods CD and 6Dj. Let n'/n be developed as a continued fraction, and let m'fm be the last convergent before n'jn, where m and mf are integers. Then n' m _ 1 n m mn' that is, mn' - m'n = 1, , 1 / . U> , . .. CD so that mco ~ mco = -(mn~ run ) = - . n x n Therefore f(z) =f(z + m'co ~ mco'), since m and m' are integers ; so that -, , ~( co\ /(*)-/(' -i- s). contravening the definition of CD as a period, viz., that no submultiple of co is a period. Hence the ratio of the periods is not a commensurable real quantity. * Ges. Werke, t. ii, pp. 25, 26. 106.] OF A UNIFORM DOUBLY-PERIODIC FUNCTION 201 If it be incommensurable, we express oj'/aj as a continued fraction. Let p/q and p'/q' be two consecutive convergents : their values are separated by the value of &>'/&>, so that we may write v~q+ \q'~q)' where 1 > h > 0. Now pq <- p'q — 1, so that - = P + — o> q qq where e is real and |e < 1 ; hence , e qa) —pa) = —, &>. Therefore f(z) =f(z + qw — pa), since p and q are integers ; so that Now since &>'/&> is incommensurable, the continued fraction is unending. We therefore take an advanced convergent, so that q' is very large. Then €- &> is a very small quantity and z + - &> is a point infinitesimally near to z, that is, the function / (V), under the present hypothesis, resumes its value at a point infinitesimally near to z. Passing along the line joining these two points infinitesimally near another, we should have / (z) constant along a line and therefore (§ 37) constant everywhere ; it would thus cease to be a varying function. The ratio of the periods is thus not an incommensurable real quantity. We therefore infer Jacobi's theorem that the ratio of the periods cannot be real. In general, the ratio is a complex quantity ; it may, however, be a pure imaginary*. COROLLARY. If a uniform function have two periods wl and &>2 such that a relation mlwl + ra2G>2 = 0 exists for integral values of m1 and ?n2, the function is only simply-periodic. And such a relation cannot exist between two periods of a simply-periodic function, if m^ and ra2 be real and incommensurable ; for then the function would be constant. * It was proved, in Ex. 5 and Ex. 6 of § 104, that certain uniform functions are doubly-periodic. A direct proof, that the ratio of the distinct periods of the functions there obtained is not a real quantity, is given by Falk, Acta Math., t. vii, (1885), pp. 197—200, and by Pringsheim, Math. Ann., t. xxvii, (1886), pp. 151—157. 202 UNIFORM [106. Similarly, if a uniform function have three periods &>1; a>.2> o>3, connected by two relations .. = 0, n1o)1 + n2a)2 + n3a)3 = 0, where the coefficients m and n are integers, then the function is only simply- periodic. 107. The two following propositions, also due to Jacobi*, are important in the theory of uniform periodic functions of a single variable : — If a uniform function have three periods w^, «2, MS such that a relation m^i + m.2&>2 + m3w3 = 0 is satisfied for integral values ofmlt w2, m3, then the function is only a doubly- periodic function. What has to be proved, in order to establish this proposition, is that two periods exist of which wl, &>2, &>3 are integral multiple combinations. Evidently we may assume that m^, ra2, m3 have no common factor: let / be the common factor (if any) of m.2 and m3, which is prime to m^. Then since and the right-hand side is an integral combination of periods, it follows that riod. is a fraction in its lowest terms. Change it into a continued -~ &>! is a period. fraction and let ^ be the last convergent before the proper value ; then 2 1 so that <l~f~P=±^f- But o>! is a period and ^ft)! is a period; therefore q —^ Wj — pwi is a period, or &>!// is a period, = to/ say. Let ra2//= m2', m3/f= m/, so that m1&V + m2'&>2 + ??i3'&)3 = 0. Change fy» m.2'/m3 into a continued fraction, taking - to be the last convergent before the proper value, so that m/ r _ 1 / i s sms * Ges. Werke, t. ii, pp. 27—32. 107.] DOUBLY-PERIODIC FUNCTIONS 203 Then r&>2 + sco., being an integral combination of periods, is a period. But ± &>2 = &)2 (sm2r — rm3) = — ra>.2m3 — s (m^ + w3'&>3) = — m^sw-i' - ma' (r&>2 + su>3) ; also + ft)3 = &)3 (sm/ — rm3) — sm2'o)3 + r (mjO)/ and o>! =/&)/. Hence two periods &>/ and r<u2 + s&>3 exist of which co1} co2, &>3 are integral multiple combinations ; and therefore all the periods are equivalent to &>/ and r&>2 + so)3, that is, the function is only doubly-periodic. COROLLARY. If a function have four periods <ul3 &>2, cos, &>4 connected by two relations m1o)1 + m2o)2 + wi3ft)3 + ra4&>4 = 0, 72J60J + W20)2 + W3ft)3 + W4«04 = 0, where the coefficients m and w are integers, the function is only doubly- periodic. 108. If a uniform function of one variable have three periods a)l, w.,, &>3, then a relation of the form m1o)l + w?2to2 + in3(i)3 = 0 must be satisfied for some integral values ofml} m2, ms. Let a)r = ar + i@r, for r = 1, 2, 3 ; in consequence of § 106, we shall assume that no one of the ratios of twj, <w2, w3 in pairs is real, for, otherwise, either the three periods reduce to two immediately, or the function is a constant. Then, determining two quantities A, and fj, by the equations so that X and //, are real quantities and neither zero nor infinity, we have for real values of X and p. Then, first, if either X or fj. be commensurable, the other is also commen surable. Let X = a/6, where a and b are integers ; then = bo)3 — aa)}, so that fyu,&>2 is a period. Now, if b/j, be not commensurable, change it into a continued fraction, and let p/q, p'/q be two consecutive convcrgents, so that, as in § 106, / P , x bfji,=^+ —,, q qq 204 TRIPLY-PERIODIC UNIFORM [108. where 1 > x > — 1. Then - &>., + -— ? is a period, and so is <w2 ; hence q qq 'P~ ^x IT is a period, that is, - <a2 is a period. We may take q indefinitely large, and then the function has an infinitesimal quantity for a period, that is, it would be a constant under the hypothesis. Hence &/* (and therefore /*) cannot be incommensurable, if X be commensurable; and thus X and //. are simul taneously commensurable or simultaneously incommensurable. CL G If X and fj, be simultaneously commensurable, let X = j- , p = -^ , so that a c &)3 = r &>! + -jG>2. o a and therefore 6rfto3 = ac^ + bca)2, a relation of the kind required. If X and //. be simultaneously incommensurable, express A, as a continued fraction ; then by taking any convergent r/s, we have r _ x *=*' /Yt where 1 > x > — 1, so that s\ — r=-: s by taking the convergent sufficiently advanced the right-hand side can be made infinitesimal. Let i\ be the nearest integer to the value of s/j,, so that, if we have A numerically less than ^. Then x sat-, — ra>1 — r1w2 = — a)1 + Aw.,, s fp and the quantity - Wj can be made so small as to be negligible. Hence S integers r, rlt s can be chosen so as to give a new period &>/(= A&>2), such that | &)/ < \ &)2 . We now take wl, &>2', &>3: they will be connected by a relation of the form 0>3 =X'(W1 +yLt/G)2/, and X' and // must be incommensurable : for otherwise the substitution for to/ of its value just obtained would lead to a relation among a>l) &>o, &>3 that would imply commensurability of X and of p. Proceeding just as before, we may similarly obtain a new period &>2" such that <o2" < \ ! mz I and so on in succession. Hence we shall obtain, after n 108.] FUNCTIONS DO NOT EXIST 205 such processes, a period co2(w) such that |&)2(n)| < ^ a>*\, so that by making n z sufficiently large we shall ultimately obtain a period less than any assigned quantity. Let such period be to ; then /(*+«)-/(*), and so for points along the co-line we have an infinite number close together at which the function is unaltered in value. The function, being uniform, must in that case be constant. It thus appears that, if A. and /j, be simultaneously incommensurable, the function is a constant. Hence the only tenable result is that A. and //. are simultaneously commensurable, and then there is a period-equation of the form m^w^ + m.2o)2 + m3o)s = 0, where m1, w2, m3 are integers. The foregoing proof is substantially due to Jacobi (I.e.). The result can be obtained from geometrical considerations by shewing that the infinite number of points, at which the function resumes its value, along a line through z parallel to the two-line will, unless the condition be satisfied, reduce to an infinite number of points in the a)1, &)2 parallelogram which will form either a continuous line or a continuous area, in either of which cases the function would be a constant. But, if the condition be satisfied, then the points along the line through z reduce to only a finite number of points. COROLLARY I. Uniform functions of a single variable cannot have three independent periods ; in other words, triply -periodic uniform functions of a single variable do not exist* ; and, a fortiori, uniform functions of a single variable with a number of independent periods greater than two do not exist. But functions involving more than one variable can have more than two periods, e.g., Abelian transcendents ; and a function of one variable, having more than two periods, is not uniform. COROLLARY II. All the periods of a uniform periodic function of a single variable reduce either to integral multiples of one period or to linear combina tions of integral multiples of two periods whose ratio is not a real quantity. 109. It is desirable to have the parallelogram, in which a doubly- periodic function is considered, as small as possible. If in the parallelogram (supposed, for convenience, to have the origin for an angular point) there be a point a)" such that /(* + »")=/(*) for all values of z, then the parallelogram can be replaced by another. * This theorem is also due to Jacobi, (I.e., p. 202, note). 206 FUNDAMENTAL PARALLELOGRAM [109. It is evident that co" is a period of the function ; hence (§ 108) we must have co" = Aco + /AW' ; and both X and /JL, which are commensurable quantities, are less than unity since the point is within the parallelogram. Moreover, co -f co' — <»", which is equal to (1 — A,) co + (1 — /"•)&>', is another point within the parallelogram; and /(* + » + «'-«")«/(*), since co, co', co" are periods. Thus there cannot be a single such point, unless X = \ = p. But the number of such points within the parallelogram must be finite ; if there were an infinite number, they would form a continuous line or a continuous area where the uniform function had an unvarying value, and consequently (§ 37) the function would have a constant value everywhere. To construct a new parallelogram when all the points are known, we first choose the series of points parallel to the co-line through the origin 0, and of that series we choose the point nearest 0, say Al. We similarly choose the point, nearest the origin, of the series of points parallel to the co-line and nearest to it after the series that includes Al} say Bl : we take OA1} OB1 as adjacent sides of the parallelogram, and these lines as the vectorial repre sentations of the periods. No point lies within this parallelogram where the function has the same value as at 0 ; hence the angular points of the original parallelograms coincide with angular points of the new parallelograms. When a parallelogram has thus been obtained, containing no internal point fl such that the function can satisfy the equation for all values of z, it is called a fundamental, or a primitive, parallelogram, : and the parallelogram of reference in subsequent investigations will be assumed to be of a fundamental character. But a fundamental parallelogram is not unique. Let co and co' be the periods for a given fundamental parallelogram, so that every other period co" is of the form Aco + //-co', where A, and /* are integers. Take any four integers a, b, c, d such that ad — lc=±l, as may be done in an infinite variety of ways ; and adopt two new periods coj and co2, such that &>! = aco + bo)', co2 = ceo + d(o'. Then the parallelogram with coj and co2 for adjacent sides is fundamental. For we have + eo = do)1 — ba>2, + co' = — ccox + aco2, and therefore any period co" = A.CO + /uco' = (\d - fie) wl + (— \b + fj.a) eo2, save as to signs of A, and /z. 109.] OF PERIODS 207 The coefficients of o^ and &)2 are integers, that is, the point <w" lies outside the new parallelogram of reference; there is therefore no point in it such that /(* + *>")=/(*), and hence the parallelogram is fundamental. COROLLARY. The aggregate of the angular points in one division of the plane into fundamental parallelograms coincides with their aggregate in any other division into fundamental parallelograms ; and all fundamental parallelograms for a given function are of the same area. The method suggested above for the construction of a fundamental parallelogram is geometrical, and it assumes a knowledge of all the points w" within a given parallelogram for which the equation/ (z -f «")=/ (z) is satisfied. Such a point o>3 within the o^, o>2 parallelogram is given by nil m2 <Bo= - (Bi -\ -- 0>9, ' m3 J m3 2 where »&1} m2, m3 are integers. We may assume that no two of these three integers have a common factor; were it otherwise, say for m^ and wi2, then, as in § 107, a submultiple of o>3 would be a period — a result which may be considered as excluded. Evidently all the points in the parallelogram are the reduced points homologous with w3, 2o>3, ...... , (m3 — 1)«3; when these are obtained, the geometrical construction is possible. The following is a simple and practicable analytical method for the construction. Change w^/rag and mz/m3 into continued fractions; and let p/q and r/s be the last convergents before the respective proper values, so that mx p e m2 r f' m3 q gm3' m3 s sm3' where e and e' are each of them +1. Let m"> n , M ml j , ^ q — =d + — , s^ = $+ — , m3 m3 m3 m3 where X and p, are taken to be less than m3, but they do not vanish because q and s are less than m3. Then 2'eo3-^w1-(9o)2 = — (/*a>2 + f»i), *a>3-ro>2 -<£<•>! = — (Xa^ + e'tOjj) ; U vn II io the left-hand sides are periods, say Qx and O2 respectively, and since /u + e is not >m3 and X + e' is not >m3, the points Q.l and Q2 determine a parallelogram smaller than the initial parallelogram. Thus are equations defining new periods Qly Q2. Moreover , . X m-. p 65 a m9 r t'o 4>-\ -- = s-^=s*-+ -, 0 + -f^ = n —? = «- + -L : m3 m3 q qms m3 2 ms * s sm3 so that, multiplying the right-hand sides together and likewise the left-hand sides, we at once see that X/i-ee' is divisible by ms if it be not zero: let X/i — ee' = wi3A. Then, as X and p are less than m3, they are greater than A; and they are prime to it, because ee' is +1. 208 MULTIPLE [109. Hence we have Aa>j = ^Q2- t'Ql, Aa>2 = XQ1- eiV Since X and /u are both greater than A, let X = X1A + X', /x = /i1A + //, where X' and /x' are <A. Then X'/*'— «' ig divisible by A if it be not zero, say X'p - ee = AA' ; then X' and p.' are >A' and are prime to it. And now A (wj — /^iO2) = /x'Q2 ~ e'^i > A (W2 "~ ^1^1) = ^-'QI ~ f®2 i and therefore, if (a1 — /^G^Qg, <B2-X1Q1 = Q4, which are periods, we have With Q3 and Q4 we can construct a parallelogram smaller than that constructed with Qj and Q2. We now have A'Q1 = fG3+//G4, A'Q.j=X'Q3 + e'fl4, that is, equations of the same form as before. We proceed thus in successive stages : each quantity A thus obtained is distinctly less than the preceding A, and so finally we shall reach a stage when the succeeding A would be unity, that is, the solution of the pair of equations then leads to periods that determine a fundamental parallelogram. It is not difficult to prove that a>lt o>2, o>3 are combinations of integral multiples of these periods. If one of the quantities, such as X'/x'-ee', be zero, then X'=/x' = l, e = e'= ±1 ; and then Q3 and O4 are identical. If e = e' = + 1, then AQ3 = Q2 - Qj , and the fundamental parallelo gram is determined by <V = QI + - (Q2 - %), G4' = Q2 - 1 (Q2 - Qt)- If f = f = -1, then AQ3 = Q2+O15 so that, as A is not unity in this case, the fundamental parallelogram is determined by Q2 and Q3. Ex. If a function be periodic in a>1? a>2, and also in <o3 where 29co = 1 7 periods for a fundamental parallelogram are QI = Scoj + 3o)2 - 8w3 , Q2' = 3 eoj + 2co2 - 5w3 , and the values of a>1} <»2, w3 in terms of O/ and Q2' are G)2 = 2-1, a>3 = 2 Q. Further discussion relating to the transformation of periods and of fundamental parallelograms will be found in Briot and Bouquet's The'orie des fonctions elliptujues, pp. 234, 235, 268—272. 110. It has been proved that uniform periodic functions of a single variable cannot have more than two periods, independent in the sense that their ratio is not a real quantity. If then a function exist, which has two periods with a real incommensurable ratio or has more than two independent periods, either it is not uniform or it is a function (whether uniform or multi form) of more variables than one. When restriction is made to uniform functions, the only alternative is that the function should depend on more than one variable. 110.] PERIODICITY 209 In the case when three periods o)l, &>2, &>3 (each of the form a-f t/3) were assigned, it was proved that the necessary condition for the existence of a uniform function of a single variable is that finite integers mly m2, m:i can be found such that ra2cr2 + m3a3 = 0, - w3/33 = 0 ; and that, if these conditions be not satisfied, then finite integers m1} m.2, ms can be found such that both Sma and 2m/8 become infinitesimally small. This theorem is purely algebraical, and is only a special case of a more general theorem as follows : Let an, or12,..., alt r+1; a21, aw,..., a2>r+1;...; arl, «.«,..., ar,r+i be r sets of real quantities such that a relation of the form wia*i + ^2^2 + . • • + nr+l <xsr+i = 0 is not satisfied among any one set. Then finite integers m^,..., mr+1 can be determined such that each of the sums j (for 5 = 1, 2,...,r) is an infinitesimally small quantity. And, a fortiori, if fewer than r sets, each containing r+1 quantities be given, the r+1 integers can be determined so as to lead to the result enunciated ; all that i is necessary for the purpose being an arbitrary assignment of sets of real | quantities necessary to make the number of sets equal to r. But the result ! is not true if more than r sets be given. We shall not give a proof of this general theorem* ; it would follow the lines of the proof in the limited case, as given in § 108. But the theorem will be used to indicate how the value of an integral with more than two periods is affected by the periodicity. Let / be the value of the integral taken along some assigned path from an initial point ZQ to a final point z\ and let the periods be &)1} &>2,..., &>r, (where r > 2), so that the general value is / + fftjcoj + m2a)., + . . . + mrwr, where mlt m2,..., mr are integers. Now if cos = as + i/3s, for s=l, 2,..., r, when it is divided into its real and its imaginary parts, then finite integers Wi, n2,..., nr can be determined such that and n-if. are both infinitesimal ; and then 2 ns is infinitesimal. But the addition of S nscos still gives a value of the integral ; hence the value can be modified * A proof will be found in Clebsch and Gordan's Theorie der Abel'schen Functioncn, § 38. F- 14 210 MULTIPLE PERIODICITY [110. by infinitesimal quantities, and the modification can be repeated indefinitely. The modifications of the value correspond to modifications of the path from ZQ to z ; and hence the integral, regarded as depending on a single variable, can be made, by modifications of the path of the variable, to assume any value. The integral, in fact, has not a definite value dependent solely upon the final value of the variable; to make the value definite, the path by which the variable passes from the lower to the upper limit must be specified. It will subsequently (§ 239) be shewn how this limitation is avoided by making the integral, regarded as a function, depend upon a proper number of independent variables — the number being greater than unity. Ex. 1. If F0 be the value of i — -, , (n integral), taken along an assigned path, Jo (\-znY and if P = 2 I1—- ^-j(# real), then the general value of the integral is I \ ^ I n where q is any integer and mp any positive or negative integer such that 2 mp = 0. P=I (Math. Trip. Part II, 1889.) Ex. 2. Prove that v= I udz, where J o is an algebraical function satisfying the equation and obtain the conditions necessary and sufficient to ensure that i) = fadz should be an algebraical function, when u is an algebraical function satisfying an equation (Liouville, Briot and Bouquet.) CHAPTER X. SIMPLY-PERIODIC AND DOUBLY-PERIODIC FUNCTIONS. 111. ONLY a few of the properties of simply-periodic functions will be given, partly because some of them are connected with Fourier's series the detailed discussion of which lies beyond our limits, and partly because, as will shortly be explained, many of them can at once be changed into properties of uniform non-periodic functions which have already been considered. When we use the graphical method of § 105, it is evident that we need consider the variation of the function within only a single band. Within that band any function must have at least one infinity, for, if it had not, it would not have an infinity anywhere in the plane and so would be a constant ; and it must have at least one zero, for, if it had not, its reciprocal, also a simply-periodic function, would not have an infinity in the band. The infinities may, of course, be accidental or essential : their character is repro duced at the homologous points in all the bands. For purposes of analytical representation, it is convenient to use a relation Ziri so that, if the point Z in its plane have R and (*) for polar coordinates, , „ Z = =— ; log R + Z7TI ft). If we take any point A in the ^-plane and a corresponding point a in the z-plane, then, as Z describes a complete circle through A with the origin as centre, z moves along a line aal} where di is a + a). A second description of the circle makes z move from ax to aa, where a2 = ax + &> • Fig. 32. and so on in succession. 14—2 212 SIMPLE PERIODICITY [111. For various descriptions, positive and negative, the point a describes a line, the inclination of which to the axis of real quantities is the argument of &>. Instead of making Z describe a circle through A, let us make it describe a part of the straight line from the origin through A, say from A, where OA = R, to C, where 00 = R'. Then z describes a line through a perpend icular to aal} and it moves to c where Similarly, if any point A' on the former circumference move radially to a point C at a distance R from the ^-origin, the corresponding z point a' moves through a distance a'c', parallel and equal to ac : and all the points c' lie on a line parallel to aa^. Repeated description of a ^-circumference with the origin as centre makes z describe the whole line cCjCo. If then a function be simply-periodic in &>, we may conveniently take any point a, and another point a^ = a + w, through a and a^ draw straight lines perpendicular to aa1} and then consider the function within this band. The aggregate of points within this band is obtained by taking (i) all points along a straight line, perpendicular to a boundary of the band, as aa^ ; (ii) the points along all straight lines, which are drawn through the points of (i) parallel to a boundary of the band. In (i), the value of z varies from 0 to co in an expression a + z, that is, in the ^-plane for a given value of R, the angle © varies from 0 to 2?r. In (ii), the value of log R varies from — oo to +00 in an expression fi\ . log R + =— w, that is, the radius R must vary from 0 to oo . 2?r Hence the band in the 0-plane and the whole of the ^-plane are made equivalent to one another by the transformation Now let z0 be any special point in the finite part of the band for a given simply-periodic function, and let Z0 be the corresponding point in the Z-planej Then for points z in the immediate vicinity of z0 and for points Z which are consequently in the immediate vicinity of Z0, we have Ziri e to where | X differs from unity only by an infinitesimal quantity. 111.] FOURIER'S THEOREM 213 If then w, a function of z, be changed into W a function of Z, the following relations subsist : — When a point ZQ is a zero of w, the corresponding point ZQ is a zero of W. When a point z0 is an accidental singularity of w, the corresponding point Z0 is an accidental singularity of W. When a point z0 is an essential singularity of w, the corresponding point Z0 is an essential singularity of W. When a point z0 is a branch- point of any order for a function w, the corresponding point Z0 is a branch-point of the same order for W. And the converses of these relations also hold. Since the character of any finite critical point for w is thus unchanged by the transformation, it is often convenient to change the variable to Z so as to let the variable range over the whole plane, in which case the theorems already proved in the preceding chapters are applicable. But special account must be taken of the point z = oo . 112. We can now apply Laurent's theorem to deduce what is practically Fourier's series, as follows. Let f(z) be a simply-periodic function having w as its period, and suppose that in a portion of the z-plane bounded by any two parallel lines, the inclina tion of which to the axis of real quantities is equal to the argument of w, the function is uniform and has no singularities; then, at points within that portion of the plane, the function can be expressed in the form of a converging 2n-2t series of positive and of negative integral powers of e "° . In figure 32, let aa^a^... and cc^... be the two lines which bound the portion of the plane : the variations of the function will all take place within that part of the portion of the plane which lies within one of the repre sentative bands, say within the band bounded by ...ac... and . ..a^...: that is, we may consider the function within the rectangle acc^a, where it has no singularities and is uniform. Now the rectangle acc^a in the 2-plane corresponds to a portion of the Z-plane which, after the preceding explanation, is bounded by two circles 2iri 2irf with the origin for common centre and of radii | e w " | and | e u ' ; and the variations of the function within the rectangle are given by the variations of a transformed function within the circular ring. The characteristics of the one function at points in the rectangle are the same as the characteristics of the other at points in the circular ring : and therefore, from the character of the assigned function, the transformed function has no singularities and it 214 FOURIER'S THEOREM [112. is uniform within the circular ring. Hence, by Laurent's Theorem (§ 28), the transformed function is expressible in the form a series which converges within the ring : and the value of the coefficient an is given by 1 tvfj Zn+* taken along any circle in the ring concentric with the boundaries. Retransforming to the variable z, the expression for the original function is 71 = + oo Zrnriz f(z) = 2 ane~^~ . 71= -00 The series converges for points within the rectangle and therefore, as it is periodic, it converges within the portion of the plane assigned. And the value of an is Zniriz (?\P *»~ d? \z) 6 az, taken along a path which is the equivalent of any circle in the ring concentric with the boundaries, that is, along any line a'c' perpendicular to the lines which bound the assigned portion of the plane. The expression of the function can evidently be changed into the form Znvi, _ 1 r -± Wj 7 where the integral is taken along the piece of a line, perpendicular to the boundaries and intercepted between them. If one of the boundaries of the portion of the plane be at infinity, (so that the periodic function has no singularities within one part of the plane), then the corresponding portion of the ^-plane is either the part within or the part without a circle, centre the origin, according as the one or the other of the boundaries is at oo . In the former case, the terms with negative indices n are absent ; in the latter, the terms with positive indices are absent. 113. On account of the consequences of the relation subsisting between the variables z and Z, many of the propositions relating to general uniform functions, as well as of those relating to multiform functions, can be changed, merely by the transformation of the variables, into propositions relating to simply-periodic functions. One such proposition occurs in the preceding section ; the following are a few others, the full development being unnecess ary here, in consequence of the foregoing remark. * The band of reference for the simply-periodic functions considered will be supposed to include the 113.] SIMPLY-PERIODIC FUNCTIONS 215 origin : and, when any point is spoken of, it is that one of the series of homologous points in the plane, which lies in the band. We know that, if a uniform function of Z have no essential singularity, then it is a rational algebraical function, which is integral if z = cc be the only accidental singularity and is meromorphic if there be accidental singu larities in the finite part of the plane ; and every such function has as many zeros as it has accidental singularities. Hence a uniform simply-periodic function with z=cc as its sole essential singularity has as many zeros as it has infinities in each band of the plane ; the number of points at which it assumes a given value is equal to the number of its zeros ; and, if the period be w, the function is a rational algebraical ZTTIZ function of e a , which is integral if all the singularities be at an infinite distance and is meromorphic if some (or all) of them be in a finite part of the plane. But any number of the zeros and any number of the infinities may be absorbed in the essential singularity at z = oo . The simplest function of Z, thus restricted to have the same number of zeros as of infinities, is one which has a single zero and a single infinity in the finite part of the plane ; the possession of a single zero and a single infinity will therefore characterise the most elementary simply-periodic function. Now, bearing in mind the relation Zniz Z=e<*, the simplest £-pomt to choose for a zero is the origin, so that Z = 1 ; and then the simplest ^-point to choose for an infinity at a finite distance is \w, (being half the period), so that Z— — \. The expression of the function in the Z-plane with 1 for a zero and — 1 for an accidental singularity is Z~l and therefore assuming as the most elementary simply-periodic function that which in the plane has a series of zeros and a series of accidental singularities all of the first order, the points of the one being midway between those of the other, its expression is A 2iriz e" -I Zniz which is a constant multiple of tan — . Since e " is a rational fractional CD function of tan — , part of the foregoing theorem can be re-stated as follows: — If the period of the function be o>, the function is a rational algebraical function of tan — . n 216 SIMPLY-PERIODIC [113. Moreover, in the general theory of uniform functions, it was found con venient to have a simple element for the construction of products, there (§ 53) called a primary factor: it was of the type ^Z-u where the function G ( -~ j could be a constant; and it had only one infinity and one zero. Hence for simply-periodic functions we may regard tan — as a typical primary factor when the number of irreducible zeros and the (equal) number of irreducible accidental singularities are finite. If these numbers should tend to an infinite limit, then an exponential factor might have to be associated with tan — ; and the function in that case might have essential singularities elsewhere than at z = oo . 114. We can now prove that every uniform function, which has no essential singularities in the finite part of the plane and is such that all its accidental singularities and its zeros are arranged in groups equal and finite in number at equal distances along directions parallel to a given direction, is a simply-periodic function. Let to be the common period of the groups of zeros and of singularities : and let the plane be divided into bands by parallel lines, perpendicular to any line representing w. Let a, b, ... be the zeros, a, /3, ... the singularities in any one band. Take a uniform function </> (z), simply-periodic in <w and having a single zero and a single singularity in the band : we might take tan — as a value of <f> (z). Then is a simply-periodic function having only a single zero, viz., z = a and a single singularity, viz., z — a. ; for as <f> {z} has only a single zero, there is only a single point for which (f>(z) = <f) (a), and a single point for which <£ (z) — $ (a). Hence is a simply-periodic function with all the zeros and with all the infinities of the given function within the band. But on account of its periodicity it has all the zeros and all the infinities of the given function over the whole plane ; hence its quotient by the given function has no zero and no singularity over the whole plane and therefore it is a constant ; that is, the given function, 114.] FUNCTIONS 217 save as to a constant factor, can be expressed in the foregoing form. It is thus a simply-periodic function. This method can evidently be used to construct simply-periodic functions, having assigned zeros and assigned singularities. Thus if a function have a + mat as its zeros and c+m'<o as its singularities, where m and m' have all integral values from — oo to +00, the simplest form is obtained by taking a constant multiple of TTZ 7T« tan tan — TTZ , TTC tan tan — Ex. Construct a function, simply-periodic in w, having zeros given by (m+^)o> and )o> and singularities by (m + i)co and (m + §) co. The irreducible zeros are ^co and f w ; the irreducible singularities are \u> and §«. Now f.TTZ \ ( TTZ , \ I tan tan ATT I I tan tan |TT I / \ <" / \ M / 7TZ \ ( TTZ „ \ tan tan JTT ] ( tan tan |TT I / \ / is evidently a function, initially satisfying the required conditions. But, as tari^r is infinite, we divide out by it and absorb it into A' as a factor ; the function then takes the form 1 + tan - 3-tan'7^ 60 We shall not consider simply-periodic functions, which have essential singularities elsewhere than at z = <x> ; adequate investigation will be found in the second part of Guichard's memoir, (I.e., p. 147). But before leaving the consideration of the present class of functions, one remark may be made. It was proved, in our earlier investigations, that uniform functions can be expressed as infinite series of functions of the variable and also as infinite products of functions of the variable. This general result is true when the functions in the series and in the products are simply-periodic in the same period. But the function, so represented, though periodic in that common period, may also have another period : and, in fact, many doubly-periodic functions of different kinds (§ 136) are often conveniently expressed as infinite converging series or infinite converging products of simply-periodic functions. Any detailed illustration of this remark belongs to the theory of elliptic functions : one simple example must suffice. , ima' Let the real part of - - be negative, and let q denote e " ; then the function being an infinite converging series of powers of the simply-periodic function e " , is finite everywhere in the plane. Evidently 6 (z) is periodic in o>, so that = 6 (z). 218 DOUBLY-PERIODIC [114. ° Again, 0(s + «»') = 2 the change in the summation so as to give $ (z) being permissible because the extreme terms for the infinite values of n can be neglected on account of the assumption with regard to q. There is thus a pseudo-periodicity for 6(z) in a period <•>'. Similarly, if 0s(z)= q* e 2J7TZ 63(z + <a') = -e " 6(z). Then 63(z) -r-d(z) is doubly-periodic in w and 2co', though constructed only from functions simply-periodic in w : it is a function with an infinite number of irreducible accidental singularities in a band. 115. We now pass to doubly-periodic functions of a single variable, the periodicity being additive. The properties, characteristic of this important class of functions, will be given in the form either of new theorems or appropriate modifications of theorems, already established ; and the develop ment adopted will follow, in a general manner, the theory given by Liouville*. It will be assumed that the functions are uniform, unless multiformity be explicitly stated, and that all the singularities in the finite part of the plane are accidental "f*. The geometrical representation of double-periodicity, explained in § 105, will be used concurrently with the analysis; and the parallelogram of periods, to which the variable argument of the function is referred, is a fundamental parallelogram (§ 109) with periods J 2co and 2&>'. An angular point £0 for the parallelogram of reference can be chosen so that neither a zero nor a pole of the function lies on the perimeter; for the number of zeros and the number of poles in any finite area must be finite, otherwise they would form a continuous line or a continuous area, or thej would be in the vicinity of an essential singularity. This choice will, ir * In his lectures of 1847, edited by Borchardt and published in Crelle, t. Ixxxviii, (1880), pp. 277 — 310. They are the basis of the researches of Briot and Bouquet, the most complet exposition of which will be found in their Theorie des fonctions elliptiques, (2nd ed.), pp. 239—280. t For doubly-periodic functions, which have essential singularities, reference should be made to Guichard's memoir, (the introductory remarks aud the third part), already quoted on p. 147, note. J The factor 2 is introduced merely for the sake of convenience. 115.] FUNCTIONS 219 general, be made ; but, in particular cases, it is convenient to have the origin as an angular point of the parallelogram and then it not infrequently occurs that a zero or a pole lies on a side or at a corner. If such a point lie on a side, the homologous point on the opposite side is assigned to the parallelogram which has that opposite side as homologous; and if it be at an angular point, the remaining angular points are assigned to the parallelograms which have them as homologous corners. The parallelogram of reference will therefore, in general, have z0, z0 + 2&>, z0 + 2&/, z0 + 2&> + 2&>' for its angular points ; but occasionally it is desirable to .take an equivalent parallelogram having z0 ± &> + &>' as its angular points. When the function is denoted by </> (2), the equations indicating the periodicity are <£ (z + 2<w) = (f> (z) = (f> (z + 2&/). 116. We now proceed to the fundamental propositions relating to doubly-periodic functions. I. Every doubly -periodic function must have zeros and infinities within the fundamental parallelogram. For the function, not being a constant, has zeros somewhere in the plane and it has infinities somewhere in the plane ; and, being doubly-periodic, it experiences within the parallelogram all the variations that it can have over the plane. COROLLARY. The function cannot be a rational integral function of z. For within a parallelogram of finite dimensions an integral function has no infinities and therefore cannot represent a doubly-periodic function. An analytical form for <j) (z) can be obtained which will put its singu larities in evidence. Let a be such a pole, of multiplicity n ; then we know that, as the function is uniform, coefficients A can be determined so that the function f(* ~ (z-a)n~ (z-a)n-1~'"~(z-a)2~ z^a is finite in the vicinity of a ; but the remaining poles of <j> (z) are singularities of this modified function. Proceeding similarly with the other singularities b, c,..., which are finite in number and each of which is finite in degree, we have coefficients A, B, C,... determined so that A^< V i? K' 9 (z) — 2, f Z T r is finite in the vicinity of every pole of <f) (z) within the parallelogram and therefore is finite everywhere within the parallelogram. Let its value be 220 PROPERTIES [116. %(X); then for points lying within the parallelogram, the function <f>(z) is expressed in the form + A* + ^ 1 1 A, T T ; 2 — a ( ft + 1 i \9 1 ' ' ' ' / z - a> ( B2 z - a)n Bm X. ' / £—6 ( 7 \ 0 ' • • • 1 / z-b)m H, _L _L #2 S±i z-h^ (z-h? ' r (z-h)1' But though <£ (^) is periodic, ^ (2?) is not periodic. It has the property of being finite everywhere within the parallelogram ; if it were periodic, it would be finite everywhere, and therefore could have only a constant value ; and then <f> (z) would be an algebraical meromorphic function, which is not periodic. The sum of the fractions in $ (z) may be called the fractional part of the function : owing to the meromorphic character of the function, it cannot be evanescent. The analytical expression can be put in the form (z - a)~n (z - 6)-™. . .(z - h)~l F(z\ where F(z) is finite everywhere within the parallelogram. If a, /3, ..., ij be all the zeros, of degrees v, p, ..., X, within the parallelogram, then F(z) = (z-a)v(z-py ...(z-^G(z\ where G (z) has no zero within the parallelogram ; and so the function can be expressed in the form (z-a)n(z-b}m...(z-h)1 G^' where G (z) has no zero and no infinity within the parallelogram or on its boundary ; and G {z) is not periodic. The order of a doubly-periodic function is the sum of the multiplicities of all the poles which the function has within a fundamental parallelogram; and, the sum being n, the function is said to be of the nth order. All these singularities are, as already remarked, accidental; it is convenient to speak of any particular singularity as simple, double, . . . according to its multiplicity. If two doubly-periodic functions u and v be such that an equation is satisfied for constant values of A, B, C, the functions are said to be equivalent to one another. Equivalent functions evidently have the same accidental singularities in the same multiplicity. II. The integral of a doubly-periodic function round the boundary of a fundamental parallelogram is zero. 116.] OF DOUBLY-PERIODIC FUNCTIONS 221 Let ABCD be a fundamental parallelogram, the boundary of it being taken so as to pass through no pole of the function. Let A be z0, B be z0+2ca, and* <= D be z0 + 2a)': then any point in AB is / ° /Q* Q, where £ is a real quantity lying between 0 and 1 ; and therefore the integral along AB is rl Any point in EG is z0 + 2<w + 2&>'£, where £ is a real quantity lying between 0 and 1 ; therefore the integral along BC is (o 'dt, o since <^> is periodic in 2&). Any point in DC is s + 2o>' + 2<wZ, where < is a real quantity lying between 0 and 1 ; therefore the integral along CD is f° J 1 2ft)' = - I J o Similarly, the integral along DA is = - I cf> Oo + 2o>'«) 2w'^. J o Hence the complete value of the integral, taken round the parallelogram, i fi = <j>(z0 Jo which ^ is manifestly zero, since each of the integrals is the integral of a continuous function. COROLLARY. Let ty(z) be any uniform function of zt not necessarily doubly-periodic, but without singularities on the boundary. Then the * The figure implies that the argument of w' is greater than the argument of w, a hypothesis which, though unimportant for the present proposition, must be taken account of hereafter (e.g., § 129). 222 INTEGRAL RESIDUE [116. integral jty (z) dz taken round the parallelogram of periods is easily seen to be n ri •^ (z(} + Scot) 2udt + I ^(z0 + 2a> + 2m't) 2a>'dt Jo J o ri ri - V (*o + 2o>' + 2a>t) 2(odt - ^ (z0 + 2to't) 2w'dt ; Jo Jo or, if we write /• ri ri then U- (2) ^ = I I/TJ (>0 + 2w't) 2m dt - ^ (z, + 2wt) 2(odt, J Jo Jo where on the left-hand side the integral is taken positively round the boundary of the parallelogram and on the right-hand side the variable t in the integrals is real. The result may also be written in the form r rD rx \-^r(z)dz=\ ^ (z) dz — I -»K (z) dz, J J A J A the integrals on the right-hand side being taken along the straight lines AD and AB respectively. Evidently the foregoing main proposition is established, when -^ (£) and T/r2 (f) vanish for all values of £. III. If a doubly -periodic function $(z) have infinities Oj, a2, ... within the parallelogram, and if Al, A2, ... be the coefficients of (z — e^)"1, (z — a^r1, . . . respectively in the fractional part of (j> (z) when it is expanded in the parallelo gram, then A1 + A2+...=0. As the function <f>(z) is uniform, the integral f(f>(z)dz is, by (§ 19, II.), the sum of the integrals round a number of curves each including one and only one of the infinities within that parallelogram. Taking the expression for (f>(z) on p. 220, the integral Amf(z — a)~mdz round the curve enclosing a is 0, if m be not unity, and is Z>jriAl, if m be unity; the integral Kmf(z — k)~mdz round that curve is 0 for all values of m and for all points k other than a ; and the integral /^ (z) dz round the curve is zero, since % (z) is uniform and finite everywhere in the vicinity of a. Hence the integral of <£ (z) round a curve enclosing c^ alone of all the infinities is Similarly the integral round a curve enclosing a.2 alone is 27riA.2; and so on, for each of the curves in succession. Hence the value of the integral round the parallelogram is 2-rnZA. 116.] OF FUNCTIONS OF THE SECOND ORDER 223 But by the preceding proposition, the value of /(/> (2) dz round the parallelo gram is zero ; and therefore This result can be expressed in the form that the sum of the residues* of a doubly -periodic function relative to a fundamental parallelogram of periods is zero. COROLLARY 1. A doubly-periodic function of the first order does not exist. Let such a function have a for its single simple infinity. Then an expression for the function within the parallelogram is A ^-a + *^> where ^ (2) is everywhere finite in the parallelogram. By the above propo sition, A vanishes ; and so the function has no infinity in the parallelogram. It therefore has no infinity anywhere in the plane, and so is merely a constant : that is, qua function of a variable, it does not exist. COROLLARY 2. Doubly-periodic functions of the second order are of two classes. As the function is of the second order, the sum of the degrees of the infinities is two. There may thus be either a single infinity of the second degree or two simple infinities. In the former case, the analytical expression of the function is where a is the infinity of the second degree and ^ (z) is holomorphic within the parallelogram. But, by the preceding proposition, A1 = 0; hence the analytical expression for a doubly-periodic function with a single irreducible infinity a of the second degree is (z - of T * v within the parallelogram. Such functions of the second order, which have only a single irreducible infinity, may be called the first class. In the latter case, the analytical expression of the function is where c, and c2 are the two simple infinities and x(z} ig finite within the parallelogram. Then See p. 42. 224 PROPERTIES OF FUNCTIONS [116. so that, if Cl = - C.2 = C, the analytical expression for a doubly-periodic function with two simple irreducible infinities a1 and «2 ig n G ( 1 1 ( - \z-a-L z - within the parallelogram. Such functions of the second order, which have two irreducible infinities, may be called the second class. COROLLARY 3. If within any parallelogram of periods a function is only of the second order, the parallelogram is fundamental. COROLLARY 4. A similar division of doubly -periodic functions of any order into classes can be effected according to the variety in the constitution of the order, the number of classes being the number of partitions of the order. The simplest class of functions of the nth order is that in which the functions have only a single irreducible infinity of the nth degree. Evi dently the analytical expression of the function within the parallelogram is G, G, Gn (z - a)2 (z - a)3 (z - a)n * ^ '' where ^ (z) is holomorphic within the parallelogram. Some of the coefficients G may vanish ; but all may not vanish, for the function would then be finite everywhere in the parallelogram. It will however be seen, from the next succeeding propositions, that the division into classes is of most importance for functions of the second jrder. IV. Two functions, which are doubly-periodic in the same periods*, and which have the same zeros and the same infinities each in the same degrees respectively, are in a constant ratio. Let <f) and ^ be the functions, having the same periods; and let a of degree v, /3 of degree fi, ... be all the irreducible zeros of <£ and T/T; arid a of degree n, b of degree m, ... be all the irreducible infinities of <f> and of ty. Then a function G (z), without zeros or infinities within the parallelogram, exists such that , , , = (z-a)v(z-py ... G _ and another function H(z), without zeros or infinities within the parallelo gram, exists such that Hence *(*)_<?(*) - Now the function on the right-hand side has no zeros in the parallelogram, for G has no zeros and H has no infinities ; and it has no infinities in the * Such functions will be called homoperiodic. 116.] OF THE SECOND ORDER 225 parallelogram, for G has no infinities and H has no zeros : hence it has neither zeros nor infinities in the parallelogram. Since it is equal to the function on the left-hand side, which is a doubly-periodic function, it has no zeros and no infinities in the whole plane ; it is therefore a constant, say A. Thus* V. Two functions of the second order, doubly -periodic in the same periods and having the same infinities, are equivalent to one another. If one of the functions be of the first class in the second order, it has one irreducible double infinity, say at a ; so that we have where %(z) is finite everywhere within the parallelogram. Then the other function also has z = a for its sole irreducible infinity and that infinity is of the second degree ; therefore we have TT where ^ (z) is finite everywhere within the parallelogram. Hence Now x and %x are finite everywhere within the parallelogram, and therefore so is H% — Gfo. But H% — Gfo, being equal to the doubly-periodic function H(j) — Gijr, is therefore doubly-periodic ; as it has no infinities within the parallelogram, it consequently can have none over the plane and therefore it is a constant, say 7. Thus proving that the functions <j> and ty are equivalent. If on the other hand one of the functions be of the second class in the second order, it has two irreducible simple infinities, say at 6 and c, so that we have where 6(z) is finite everywhere within the parallelogram. Then the other function also has z = b and z = c for its irreducible infinities, each of them being simple ; therefore we have where 6l (z) is finite everywhere within the parallelogram. Hence (z) - Cty (z) = De (z) - Cei (z}. * This proposition is the modified form of the proposition of § 52, when the generalising exponential factor has been determined so as to admit of the periodicity. F. 15 226 IRREDUCIBLE ZEROS [116. The right-hand side, being finite everywhere in the parallelogram, and equal to the left-hand side which is a doubly-periodic function, is finite everywhere in the plane ; it is therefore a constant, say B, so that proving that <£ and ty are equivalent to one another. It thus appears that in considering doubly-periodic functions of the second order, homoperiodic functions of the same class are equivalent to one another if they have the same infinities ; so that, practically, it is by their infinities that homoperiodic functions of the second order and the same class are dis criminated. COROLLARY 1. If two equivalent functions of tlie second order have one zero the same, all their zeros are the same. For in the one class the constant /, and in the other class the constant B, is seen to vanish on substituting for z the common zero ; and then the two functions always vanish together. COROLLARY 2. If two functions, doubly-periodic in the same periods but not necessarily of the second order, have the same infinities occurring in such a, j way that the fractional parts of the two functions are the same except as to a constant factor, the functions are equivalent to one another. And if, in addition, they have one zero common, then all their zeros are common, so that the functions are then in a constant ratio. COROLLARY 3. If two functions of the second order, doubly-periodic in( the same periods, have their zeros the same, and one infinity common, they are ^ in a constant ratio. VI. Every doubly -periodic function has as many irreducible zeros as it has irreducible infinities. Let <£ (z) be such a function. Then z +h — z is a doubly-periodic function for any value of h, for the numerator is doubly- periodic and the denominator does not involve z ; so that, in the limit when h = 0, the function is doubly-periodic, that is, </>' (z) is doubly-periodic. Now suppose <f>(z) has irreducible zeros of degree m1 at a1} ra2 at a2, ..., and has irreducible infinities of degree /^ at «1} yu,2 at «2, ... ; so that the number of irreducible zeros is Wj + ra2 + . . . , and the number of irreducible infinities is ^1 + /i2 + ..., both of these numbers being finite. It has been shewn that <£ {z) can be expressed in the form 116.] AND IRREDUCIBLE INFINITIES 227 whore F(z) has neither a zero nor an infinity within, or on the boundary of, the parallelogram of reference. Since F(z) has a value, which is finite, continuous and different from zero Tjlt / \ everywhere within the parallelogram or on its boundary, the function -p4-r * W is not infinite within the same limits. Hence we have rr - ~ — ... 9 (z) z—a± z — «2 + -* + =*. + .. z — ttj z — a2 where g (z) has no infinities within, or on the boundary of, the parallelogram of reference. But, because <f> (z) and <f> (z) are doubly-periodic, their quotient is also doubly-periodic ; and therefore, applying Prop. II., we have m^ + w2 + . . . — ^ — p2 — . . . = 0, that is, m1+m2 + ... = fj,! + fi2+ ..., or the number of irreducible zeros is equal to the number of irreducible infinities. COROLLARY I. The number of irreducible points for which a doubly - periodic function assumes a given value is equal to the number of irreducible zeros. For if the value be A, every infinity of $(z) is an infinity of the doubly- periodic function $ (z) — A ; hence the number of the irreducible zeros of the latter is equal to the number of its irreducible infinities, which is the same as the number for <£ (z} and therefore the same as the number of irreducible zeros of <£ (z). And every irreducible zero of <£ (z} — A is an irreducible point, for which <£ (z) assumes the value A. COROLLARY II. A doubly-periodic function with only a single zero does not exist; a doubly -periodic function of the second order has two zeros; and, generally, the order of a function can be measured by its number of irreducible zeros. Note. It may here be remarked that the doubly-periodic functions (§ 115), that have only accidental singularities in the finite part of the plane, have z = oo for an essential singularity. It is evident that for infinite values of z, the finite magnitude of the parallelogram of periods is not recognisable ; and thus for z = GO the function can have any value, shewing that z = oo is an essential singularity. VII. Let a1} a2)... be the irreducible zeros of a function of degrees w1; m2, ... respectively ; a1} «2, ... its irreducible infinities of degrees /^, /u,2, ... respectively; and z1,z2,... the irreducible points where it assumes a value c, which is neither zero nor infinity, their degrees being M1} M.2) ... respectively. 15—2 228 IRREDUCIBLE ZEROS [116. Then, except possibly as to additive multiples of Hie periods, the quantities 2 mrar, 2 UrCir and 2 Mrzr are equal to one another, so that r=l r=l r=l 2 mrar = 2 Mrzr = 2 prctr (mod. 2o>, 2&/)- r=l r=l r=l Let (/> (/) be the function. Then the quantities which occur are the sums of the zeros, the assigned values, and the infinities, the degree of each being taken account of when there is multiple occurrence ; and by the last proposition these degrees satisfy the relations The function <f)(z) — c is doubly-periodic in 2«u and 2&>' ; its zeros are z1} z.2, ... of degrees M1} M^,... respectively; and its infinities are ctl, «2, ... of degrees /i1} yn2, •••, being the same as those of <£(Y). Hence there exists a function G(z), without either a zero or an infinity lying in the parallelogram or on its boundary, such that </> 0) - c can be expressed in the form ^l*1C.(*I*a>!'" G (*) for all points not outside the parallelogram ; and therefore, for points in that region <f>'(Y) ^ Mr ^ *r G'(z) \ /~** / \ • / \ <j)(z) — C r=l Z — Zr Z— O.r Hence z$(z) ~ Mrz v prz zG' (z) . . >. - — 2< - --- 2* --- 1 — .~ , . $(z) — C r=l z — zr Z— ar W (*) = 2 Jfr+ 2 , ~r * ~r /-v / -. , =\Z— Zr Z—OLr (r(z) 2 Mr= 2 nr. r=l r=l Integrate both sides round the boundary of the fundamental parallelogram. Because G (z) has no zero and no infinity in the included region and does not vanish along the curve, the integral 'zG'(z) I dz G(z) vanishes. But the points z{ and 04 are enclosed in the area ; and therefore the value of the right-hand side is 2iri 2 Mrzr — Ziri 2 /V*r, so that \Z) — c the integral being extended round the parallelogram. 116.] AND IRREDUCIBLE INFINITIES 229 zd>' (z) Denoting the subject of integration , by/(^), we have <p(z) — c -/«=*" - and therefore, by the Corollary to Prop. II., the value of the foregoing integral is *• r £¥-*-** r £¥-*• JA<f>(Z)-C JA(j)(z)-G the integrals being taken along the straight lines AD and AB respectively (fig. 33, p. 221). Let w — <f)(z) — c; then, as z describes a path, w will also describe a single path as it is a uniform function of z. When z moves from A to D, w moves from (j>(A)-c by some path to (f>(D) — c, that is, it returns to its initial position since <f> (D) = <f> (A) ; hence, as z describes AD, w describes a simple closed path, the area included by which may or may not contain zeros and infinities of w. Now dw = <f>' (z) dz, CD <£' (z\ and therefore the integral I ,,\ dz is equal to * JAJ>(*)-C I dw w taken in some direction round the corresponding closed path for w. This integral vanishes, if no w-zero or w-infinity be included within the area bounded by the path ; it is + Im'iri, if m be the excess of the number of included zeros over the number of included infinities, the + or — sign being taken with a positive or a negative description ; hence we have where m is some positive or negative integer and may be zero. Similarly where n is some positive or negative integer and may be zero. Thus 27Ti (2,MrZr ~ 2/V*,-) = 2w . 2w7n — 2a)' . Smri, and therefore ^Mrzr — ^prir = 2ma> — 2?io>' = 0 (mod. 2&), 2o>'). Finally, since ^Mrzr = 2/v*r whatever be the value of c, for the right-hand 230 DOUBLY-PERIODIC FUNCTIONS [116. side is independent of c, we may assign to c any value we please. Let the value zero be assigned ; then ^Mrzr becomes Smrar, so that ^mrar = "2/j,rf*r (mod. 2&>, 2&/). The combination of these results leads to the required theorem*, expressed by the congruences 2 mrar = 2 Mrzr = 2 ^r^r (mod. 2o>, 2&>'). r=l r=l r=l Note. Any point within the parallelogram can be represented in the form z0 + a2&> + 62&>', where a and 6 are real positive quantities less than unity. Hence 2 Mrzr = Az'2a> + Bz2a>/ + z£Mr, where J. and B are real positive quantities each less than 27lfr, that is, less than the order of the function. In particular, for functions of the second order, we have z1 + z, = Az 2&> + Bz 2&/ + 2.2-0, where Az and Bz are positive quantities each less than 2. Similarly, if a and b be the zeros, a + b = Aa 2w + £a 2w' + 2*o, where J.ffl and Ba are each less than 2 ; hence, if ^i + ^2 — a — b — w2w + m'2o>', then w may have any one of the three values - 1, 0, 1 and so may m', the simultaneous values not being necessarily the same. Let a and ft be the infinities of a function of the second class ; then a + /3 — a — b = ?i2&) + n"2w', where n and ri may each have any one of the three values — 1, 0, 1. By changing the origin of the fundamental parallelogram, so as to obtain a different set of irreducible points, we can secure that n and n' are zero, and then a + @ = a+b. Thus, if n be 1 with an initial parallelogram, so that a + /3 = a + &+2&>, we should take either /3 - 2&> = {¥, or a - 2&> = a', according to the position of a and /3, and then have a new parallelogram such that a + @' = a + b, or a' + ft = a + b. The case of exception is when the function is of the first class and has a repeated zero. * The foregoing proof is suggested by Konigsberger, Theorie der elliptischen Functionen, t. i, p. 342 ; other proofs are given by Briot and Bouquet and by Liouville, to whom the adopted form of the theorem is due. The theorem is substantially contained in one of Abel's general theorems in the comparison of transcendents. 116.] OF THE SECOND ORDER 231 VIII. Let $ (z) be a doubly -periodic function of the second order. If 7 be the one double infinity when the function is of the first class, and if a and ft be the two simple infinities when the function is of the second class, then in the former case and in the latter case </> (z) — <£ (a + (3 — z). Since the function is of the second order, so that it has two irreducible infinities, there are two (and only two) irreducible points in a fundamental parallelogram at which the function can assume any the same value : let them be z and z'. Then, for the first class of functions, we have z + z' = 27 = 27 + 2mo> + 2wa>', where m and n are integers ; and then, since <f)(z) = <j> (z'} by definition of z and /, we have <£ (z) = <£ (27 - z + 2ma) = 0(27-4 For the second class of functions, we have z + z = a. + /3 so that, as before, (/> (z) = </> (a + /3 - z + 2ma) + 2wa>') 117. Among the functions which have the same periodicity as a given function </> (z), the one which is most closely related to it is its derivative <£' (z). We proceed to find the zeros and the infinities of the derivative of a function, in particular, of a function of the second order. Since (f> (z) is uniform, an irreducible infinity of degree n for </> (z) is an irreducible infinity of degree n -f 1 for §' (z). Moreover <£' (z), being uniform, has no infinity which is not an infinity of </> (z) ; thus the order of <£' (z) is 2(?i + l) or its order is greater than that of cj>(z) by an integer which represents the number of distinct irreducible infinities of <£ (z), no account being taken of their degree. If, then, a function be of order m, the order of its derivative is not less than m + 1 and is not greater than 2m. Functions of the second order either possess one double infinity so that within the parallelogram they take the form — and then <j>' (z) = — - — + %' (*), 232 ZEROS OF THE DERIVATIVE [117. that is, the infinity of (f>(z) is the single infinity of tf>' ' (z) and it is of the third degree, so that cf>' (z) is of the third order ; or they possess two simple infinities, so that within the parallelogram they take the form and then f W = - G - - _ + x' (,), that is, each of the simple infinities of <£ (z) is an infinity for </>' (z) of the second degree, so that <£' (z) is of the fourth order. It is of importance (as will be seen presently) to know the zeros of the derivative of a function of the second order. For a function of the first class, let 7 be the irreducible infinity of the second degree ; then we have and therefore $'(2) = — </>' (^7 — z). Now </>' (z) is of the third order, having 7 for its irreducible infinity in the third degree : hence it has three irreducible zeros. In the foregoing equation, take z = 7 : then </>' (7) = -$' (7), shewing that 7 is either a zero or an infinity. It is known to be the only infinity of <£' (z). Next, take z = 7 + &> ; then <£' (7 + &)) = — $' (7 — a>) = - <£' (7 + G>), shewing that 7 + &> is either a zero or an infinity. It is known not to be an infinity ; hence it is a zero. Similarly 7 + &/ and 7 + <u + &/ are zeros. Thus three zeros are obtained, distinct from one another ; and only three zeros are required ; if they be not within the parallelogram, we take the irreducible points homologous with them. Hence : IX. The three zeros of the derivative of a function, doubly -periodic in 2eo and 2eo' and having 7 for its double (and only) irreducible infinity, are 7 + &), 7 + eo', 7 + w + w . For a function of the second class, let a and /3 be the two simple irreducible infinities; then we have and therefore <f>' (z)= — <f>' (a + ft — z). 117.] OF A DOUBLY-PERIODIC FUNCTION 233 Now (j) (z) is of the fourth order, having a and ft as its irreducible infinities each in the second degree ; hence it must have four irreducible zeros. In the foregoing equation, take z = \(VL + ft) ; then shewing that | (a + /3) is either a zero or an infinity. It is known not to be an infinity ; hence it is a zero. Next, take z = £ (a + (3) + w ; then f(}(«t£)+«} --+'{*(«+£)-••] = - <£' & (a + £) - to + 2&>j —.+'{*<«+£)+••}, shewing that |(a + /3) + &> is either a zero or an infinity. As before, it is a zero. Similarly i (a + /3) + &>' and i (a + /3) -f &> + &>' are zeros. Four zeros are thus obtained, distinct from one another; and only four zeros are required. Hence : X. The four zeros of the derivative of a function, doubly-periodic in 2&> and 2o)' and having a and /3 for its simple (and only) irreducible infinities, are i(a + /3), i(a + /3) + a>, i(a + /3) + ft>', |- (a + /3) + w + a/. The verification in each of these two cases of Prop. VII., that the sum of the zeros of the doubly-periodic function <£' (z) is congruent with the sum of its infinities, is immediate. Lastly, it may be noted that, if zl and z^ be the two irreducible points for which a doubly -periodic function of the second order assumes a given value, then the values of its derivative for z1 and for z% are equal and opposite. For (j> (z) = <f> (a + /3 - z) = cf> (z, + z.2 - z), since zl + z., = a + (3 ; and therefore <f> (z) = -$' (z, + z.2- z), that is, <£' (zl) = — </>' (z2), which proves the statement. 118. We now come to a different class of theorems. XI. Any doubly -periodic function of the second order can be expressed algebraically in terms of an assigned doubly-periodic function of the second order, if the periods be the same. The theorem will be sufficiently illustrated and the line of proof sufficiently indicated, if we express a function (/> (z) of the second class, with irreducible infinities a, ft and irreducible zeros a, b such that a + (3 = a + b, in 234 FUNCTIONS [118. terms of a function <£ of the first class with 7 as its irreducible double infinity. n .. , ,. Consider a function Q (z + h) _ A zero of <X> (z + h) is neither a zero nor an infinity of this function ; nor is an infinity of <1> (z + h) a zero or an infinity of the function. It will have a and 6 for its irreducible zeros, if a + h = h', b + h + h' = 27 ; and these will be the only zeros, for <E> is of the second order. It will have o and yS for its irreducible infinities, if and these will be the only infinities, for <£ is of the second order. These equations are satisfied by Hence the assigned function, with these values of h, has the same zeros and the same infinities as $>(z); and it is doubly-periodic in the same periods. The ratio of the two functions is therefore a constant, by Prop. IV., so that c|> (z + h) — <I> (h') If the expression be required in terms of <& (z) alone and constants, then <j> (z 4. h} must be expressed in terms of <I> (z) and constants which are values of <X> (z) for special values of z. This will be effected later. The preceding proposition is a special case of a more general theorem which will be considered later ; the following is another special case of that theorem : viz. : XII. A doubly -periodic function with any number of simple infinities can be expressed either as a sum or as a product, of functions of the second order and the second class which are doubly-periodic in the same periods. Let «j, «2, ..., an be the irreducible infinities of the function <£, and suppose that the fractional part of <t> (z) is •A-i , A2 [ ^ i-+ ^n z — ttj z — «2 z — an ' with the condition A1 + A2 + + An = Q. Let <j>n(z) be a function, doubly-periodic in the same periods, with a,-, a,- as its only irreducible infinities, 118.] OF THE SECOND ORDER 235 supposed simple; where i and j have the values 1, ,n. Then the fractional parts of the functions ^>j, (z), <£23 (z), . . . are 0, G, i z — a., I \z — «2 ^ — «, respectively; and therefore the fractional part of ^!^ / \ , • 0» W + is Al An- An~l z-a.! Z-CL, z-cin-T. z-an •Ai An_^ An = - -+...+- - + — ^, Z-Cl! Z- «„_! Z - Ctn n since S -4* = 0. This is the same as the fractional part of <l> (z); and therefore - <^>23 (f) - ... - -~ has no fractional part. It thus has no infinity within the parallelogram ; it is a doubly-periodic function and therefore has no infinity anywhere in the plane; and it is therefore merely a constant, say B. Hence, changing the constants, we have $>(z)-B^(z}-B.><t>v(z)-...-Bn-,<t>n-,,n(z} = B, giving an expression for <$> (z} as a linear combination of functions of the second order and the second class. But as the assignment of the infinities is arbitrary, the expression is not unique. For the expression in the form of a product, we may denote the n irreducible zeros, supposed simple, by «!,...,«„. We determine n - 2 new irreducible quantities c, such that C2= Cn—2 — &n—\ ~r Cn—3 ~ Q"n—i > Cln = ttn + C_ — Q"— n this being possible because 2 o^ = 2 ar ; and we denote by $ (z ; a, ft ; e, f) a »•=! r=l function of .gr, which is doubly-periodic in the periods of the given function, ALGEBRAICAL RELATIONS [118. has a and $ for simple irreducible infinities and has e and / for simple irreducible zeros. Then the function <f)(z; al5 «2 ; «i, Ci) <f> (z ; as, ci ; 0-2, c2) ...<£ (2 ; «n, cn_2 ; an_l5 an) has neither a zero nor an infinity at c1} at c2, ..., and at cn_2 ; it has simple infinities at al} a2, ..., an, and simple zeros at alt a2, ..., an-1} an. Hence it has the same irreducible infinities and the same irreducible zeros in the same degree as the given function <£ (z) ; and therefore, by Prop. IV., <I> (z) is a mere constant multiple of the foregoing product. The theorem is thus completely proved. Other developments for functions, the infinities of which are not simple, are possible ; but they are relatively unimportant in view of a theorem, Prop. XV., about to be proved, which expresses any periodic function in terms of a single function of the second order and its derivative. XIII. If two doubly -periodic functions have the same periods, they are connected by an algebraical equation. Let u be one of the functions, having n irreducible infinities, and v be the other, having m irreducible infinities. By Prop. VI., Corollary I., there are n irreducible values of z for a value of u; and to each irreducible value of z there is a doubly-infinite series of. values of z over the plane. The function v has the same value for all the points in any one series, so that a single value of v can be associated uniquely with each of the irreducible values of z, that is, there are n values of v for each value of u. Hence, (§ 99), v is a root of an algebraical equation of the nth degree, the coefficients of which are functions of u. Similarly u is a root of an algebraical equation of the mth degree, the coefficients of which are functions of v. Hence, combining these results, we have an algebraical equation between u and v of the nth degree in v and the mth in u, where m and n are the respective orders of v and u. COROLLARY I. If both the functions be even functions of z, then n and m are even integers ; and the algebraical relation between u and v is of degree ^n in v and of degree ^m in u. COROLLARY II. If a function u be doubly-periodic in &> and &>', and a function v be doubly -periodic in fl and U', where n = mca + nta, I!' = m'w + nw! , m, n, m', n being integers, then there is an algebraic relation between u and v. 119. It has been proved that, if a doubly-periodic function u be of order m, then its derivative du/dz is doubly-periodic in the same periods and is of an order n, which is not less than m + 1 and not greater than 2?/i. Hence, by 119.] BETWEEN HOMOPERIODIC FUNCTIONS 237 Prop. XIII., there subsists between u and u an algebraical equation of order m in u' and of order n in u; let it be arranged in powers of u' so that it takes the form U" u'm _j_ JJ u'm—i _i _ _ i U _ u'2 i JJ _u' i JJ __ Q where U0, U1} ... , Um are rational integral algebraical functions of u one at least of which must be of degree n. Because the only distinct infinities of u' are infinities of u, it is impossible that u' should become infinite for finite values of u: hence U0 = 0 can have no finite roots for u, that is, it is a constant and so it may be taken as unity. And because the m values of z, for which u assumes a given value, have their sum constant save as to integral multiples of the periods, we have corresponding to a variation 8u ; or du du du f/7/ Now — is one of the values of u' corresponding to the value of u, and so for the others ; hence 3 1 r=i ur that is, by the foregoing equation, " m— i = 0, un. and therefore Um-^ vanishes. Hence : XIV. There is a relation, between a doubly -periodic function u of order m and its derivative, of the form u'm + U^'™-1 + ...+ U^u'* + Um = 0, where Ul}..., Um_2, Um are rational integral algebraical functions of u, at least one of which must be of degree n, the order of the derivative, and n is not less than m + 1 and not greater than 2m. Further, by taking v = - , which is a function of order m because it has the Uj m irreducible zeros of u for its infinities, and substituting, we have vf™ _ 03 U^'m~l + v*U«v'm~* - . . . ± v2"1-4 Um_2v''2 + v2"1 Um = 0. The coefficients of this equation must be integral functions of v ; hence the degree of Ur in u cannot be greater than 2r. COROLLARY. The foregoing equation becomes very simple in the case of doubly-periodic functions of the second order. Then m = 2. 238 DIFFERENTIAL EQUATION [119. If the function have one infinity of the second degree, its derivative has that infinity in the third degree, and is of the third order, so that n = 3 ; and the equation is /y7?/\2 ( ^ ) = \u? + 3/iw2 + Svu + p, \d*J where X, /*, v, p are constants. If 6 be the infinity, so that A *.£(,)_-_— + £(*), where % (^) is everywhere finite in the parallelogram, then - = ±A ; and the /77/ zeros of -j- are 6 + o>, 0 + &/, 6 + o> + CD' ; so that diz a,')} { This is £/ie general differential equation of Weierstrasss elliptic functions. If the function have two simple infinities a and @, its derivative has each of them as an infinity of the second degree, and is of the fourth order, so that n = 4 ; and the equation is (du\* _ (dz) = dM + c2w + >c3u + c4, where c0, c1} c2, c3, c4 are constants. Moreover where ^ (^) is finite everywhere in the parallelogram. Then cu = G~2 ; and ^/'i/ the zeros of -y- are ^ (a + /3), -|- (a + (3) + w, ^ (a -f /3) + cof, % (a + ft) + w + &>', ft/2 so that the equation is (« + 13)+ « + «}]. This is the general differential equation of Jacobis elliptic functions. The canonical forms of both of these equations will be obtained in Chapter XI., where some properties of the functions are investigated as special illustra tions of the general theorems. Note. All the derivatives of a doubly-periodic function are doubly- periodic in the same periods, and have the same infinities as the function but in different degrees. In the case of a function of the second order, which must satisfy one or other of the two foregoing equations, it is easy to see that a derivative of even rank is a rational, integral, algebraical function of u, and that a derivative of odd rank is the product of a rational, integral, algebraical function of u by the first derivative of u. 119.] OF DOUBLY-PERIODIC FUNCTIONS 239 It may be remarked that the form of these equations confirms the result at the end of § 117, by giving two values of u' for one value of u, the two values being equal and opposite. Ex. If u be a doubly-periodic function having a single irreducible infinity of the third degree so as to be expressible in the form 2 6 — -o + -5 + integral function of z z z within the parallelogram of periods, then the differential equation of the first order which determines u is where £74 is a quartic function of u and where a is a constant which does not vanish with 6. (Math. Trip., Part II, 1889.) XV. Every doubly -periodic function can be expressed rationally in terms of a function of the second order, doubly-periodic in the same periods, and its derivative. Let u be a function of the second order and the second class, having the same two periods as v, a function of the rath order ; then, by Prop. XIII., there is an algebraical relation between u and v which, being of the second degree in v and the mth degree in u, may be taken in the form Lv* - 2Mv + P = 0, where the quantities L, M, P are rational, integral, algebraical functions of u and at least one of them is of degree m. Taking Lv-M=w, we have w2 = M'2 — LP, a rational, integral, algebraical function of u of degree not higher than 2w. Thus w cannot be infinite for any finite value of u : an infinite value of u makes w infinite, of finite multiplicity. To each value of u there correspond two values of w equal to one another but opposite in sign. Moreover w, being equal to Lv - M , is a uniform function of z, say F(z\ while it is a two-valued function of u. A value of u gives two distinct values of z, say zl and £2 ; hence the values of w, which arise from an assigned value of u, are values of w arising as uniform functions of the two distinct values of z. Hence as the two values of w are equal in magnitude and opposite in sign, we have r(4)+J*(4)-Oi that is, since ^ + z.2 = a. + ft where a and /3 are the irreducible infinities of u, so that l(a + £), £(a + /3) + a>, £(a + £) + «', and £ (a + /3) + a> + a>' are either zeros or infinities of w. They are known not to be infinities of u, and w is infinite only for infinite values of u ; hence the four points are zeros of w. 240 RELATIONS BETWEEN [119. But these are all the irreducible zeros of u' ; hence the zeros of u' are included among the zeros of w. Now consider the function w/u'. The numerator has two values equal and opposite for an assigned value of u ; so also has the denominator. Hence w/u' is a uniform function of u. This uniform function of u may become infinite for (i) infinities of the numerator, (ii) zeros of the denominator. But, so far as concerns (ii), we know that the four irreducible zeros of the denominator are all simple zeros of u' and each of them is a zero of w .; hence w/u' does not become infinite for any of the points in (ii). And, so far as concerns (i), we know that all of them are infinities of u. Hence w/u, a uniform function of u, can become infinite only for an infinite value of u, and its multiplicity for such a value is finite; hence it is a rational, integral, algebraical function of u, say N, so that w = Nu'. Moreover, because w2 is of degree in u not higher than 2m, and u'2 is of the fourth degree in u, it follows that N is of degree not higher than m — 2. We thus have Lv — M — Nu, M+Nu M N , v= ~r = L + LU> where L, M, N are rational, integral, algebraical functions of u ; the degrees of L and M are not higher than m, and that of N is not higher than m — 2. Note 1. The function u, which has been considered in the preceding proof, is of the second order and the second class. If a function u of the second order and the first class, having a double irreducible infinity, be chosen, the course of proof is similar ; the function w has the three irreducible zeros of u' among its zeros and the result, as before, is w = Nu'. But, now, w"- is of degree in u not higher than 2m and u'2 is of the third degree in u ; hence N is of degree not higher than m — 2 and the degree of w2 in u cannot be higher than 2m — 1. Hence, if L, M, P be all of degree m, the terms of degree 2m in LP — M2 disappear. If all of them be not of degree m, the degree of M must not be higher than m — l ; the degree of either L or P must be m, but the degree of the other must not be greater than m—l, for otherwise the algebraical equation between u and v would not be of degree m in u. We thus have Lv2 - 2Mv + P = (), Lv - M = Nu', 119.] HOMOPERIODIC FUNCTIONS 241 where the degree of N in u is not higher than m — 2. If the degree of L be less than TO, the degree of M is not higher than TO — 1 and the degree of P is TO. If the degree of L be m, the degree of M may also be m provided that the degree of P be TO and that the highest terms be such that the coefficient of u2m in LP - M'2 vanishes. Note 2. The theorem expresses a function v rationally in terms of u and u : but u' is an irrational function of u, so that v is not expressed rationally in terms of u alone. But, in Propositions XI. and XII., it was indicated that a function such as v could be rationally expressed in terms of a doubly-periodic function, such as u. The apparent contradiction is explained by the fact that, in the earlier propositions, the arguments of the function u in the rational expression and of the function v are not the same ; whereas, in the later proposition whereby v is expressed in general irrationally in terms of u, the arguments are the same. The transition from the first (which is the less useful form) to the second is made by expressing the functions of those different arguments in terms of functions of the same argument when (as will appear subsequently, in § 121, in proving the so-called addition-theorem) the irrational function of u, represented by the derivative u, is introduced. COROLLARY I. Let H denote the sum of the irreducible infinities or of the irreducible zeros of the function u of the second order, so that H = 2y for functions of the first class, and O = a + /3 for functions of the second class. Let u be represented by <f> (z) and v by ty (z), when the argument must be put in evidence. Then so that W-Z) = J-j i_j ±j Hence ^ (z) + ^ (fl - z) = 2 ^= 2R, JL First, if y (z) = ,Jr (ft - z\ then S = 0 and ^ (z) = R : that is, a function ^ (z), which satisfies the equation can be expressed as a rational algebraical meromorphic function of <f> (z) of the second order, doubly -periodic in the same periods and having the sum of its irreducible infinities congruent with O. Second, if ^ (e) = - y, (fl _ z\ then R = 0 and ^ (*) = flf (*) ; that is, function ^ (z), which satisfies the equation 16 a 242 HOMOPERIODIC FUNCTIONS [119. can be expressed as a rational algebraical meromorphic function of <£ (z), multiplied by 0' (z), where $ (z} is doubly-periodic in the same periods, is of the second order, and has the sum of its irreducible infinities congruent with Q. Third, if ty(z) have no infinities except those of u, it cannot become infinite for finite values of u ; hence L = 0 has no roots, that is, L is a constant which may be taken to be unity. Then i/r (z) a function of order m can be expressed in the form where, if the function </> (z) be of the second class, the degree of M is not higher than m ; but, if it be of the first class, the degree of M is not higher than m - 1 ; and in each case the degree of N is not higher than m - 2. It will be found in practice, with functions of the first class, that these upper limits for degrees can be considerably reduced by counting the degrees of the infinities in Thus, if the degree of M in u be ^ and of N be \ the highest degree of an : infinity is either 2/t or 2X + 3 ; so that, if the order of ^ (z) be m, we should have m = 2/j, or m = 2\ + 3, > according as m is even or odd. When functions of the second class are used to represent a function ^r (z), which has two infinities a and /3 each of degree n, then it is easy to see that M is of degree n and N of degree n - 2 ; and so for other cases. COROLLARY II. Any doubly -periodic function can be expressed rationally in terms of any other function u of any order n, doubly-periodic in the same periods, and of its derivative ; and this rational expression can always be taken in the form U0 + U,U' + t/X3 + • • • + Un-,u'n~\ where U0, ... , £7n-i are algebraical, rational, meromorphic functions of u. COROLLARY III. If <f) be a doubly-periodic function, then <f> (u + v) can be expressed in the form where ^ is a doubly -periodic function in the same periods and of the second order : each of the functions A, D, E is a symmetric function of^(u) and i/r (v), and B is the same function of^(v) and ty(u) as C is of ty (u) and ty (v). The degrees of A and E are not greater than m in ty (u) and than m in ^ (v), where m is the order of </> ; the degree of D is not greater than m - 2 in ^ (u) and than m - 2 in ^ (v) ; the degree of B is not greater than m - 2 in ^ (u) and than m in ^ (v), and the degree of C is not greater than m - 2 in -^ (v) and than m in -^ (u). CHAPTER XI. DOUBLY-PERIODIC FUNCTIONS OF THE SECOND ORDER. THE present chapter will be devoted, in illustration of the preceding theorems, to the establishment of some of the fundamental formulae relating to doubly-periodic functions of the second order which, as has already (in § 119, Cor. to Prop. XIV.) been indicated, are substantially elliptic functions : but for any development of their properties, recourse must be had to treatises on elliptic functions. It may be remarked that, in dealing with doubly-periodic functions, we may restrict ourselves to a discussion of even functions and of odd functions. For, if (/> (z) be any function, then £ {<j> (z} + <j>(— z}} is an even function, and \ {(f>(z) — </>(— z}} is an odd function, both of them being doubly-periodic in the periods of <f> (z) ; and the new functions would, in general, be of order double that of <J>(z). We shall practically limit the discussion to even functions and odd functions of the second order. 120. Consider a function <j>(z\ doubly-periodic in 2&> and 2w'; and let it be an odd function of the second class, with a and ft as its irreducible infinities, and a and b as its irreducible zeros*. Then we have <£ (z) = (f> (a + /3 — z) which always holds, and <f> (— z) = — </> (z) which holds because <£ (z) is an odd function. Hence <f> (a + /3 + z) = (/>(- *) = -$(*) so that a + ft is not a period ; and -*(*), To fix the ideas, it will be convenient to compare it with snz, for which 2w = 4^T, 2<a' = a=iK', p=iK' + 2K, a-0, and b = 2K. 16—2 244 DOUBLY-PERIODIC FUNCTIONS [120. whence 2 (a + /S) is a period. Since a -f /3 is not a period, we take a + /3 = a>, or = &)', or = &> + w' ; the first two alternatives merely interchange &> and &>', so that we have either a + /3 = o), or a + /3 = ft) + &>'. And we know that, in general, a + b = a + /3. First, for the zeros : we have so that </>(0) is either zero or infinite. The choice is at our disposal; for - satisfies all the equations which have been satisfied by $(z) and an </>(*) infinity of either is a zero of the other. We therefore take so that we have a = 0, 6 = to or &) + ft)'. Next, for the infinities : we have *(*)—$(-*) and therefore <j> (- a) = - $ (a) = oo . The only infinities of <£ are a and /3, so that either — a= a, or -CL = P. The latter cannot hold, because it would give a + /3 = 0 whereas or = &> + &/; hence 2a = 0, which must be associated with a + /3 = w or with a + /3 = &> + &/. Hence a, being a point inside the fundamental parallelogram, is either 0, a), &)', or tw + &)'. It cannot be 0 in any case, for that is a zero. If a _|_ ^ = Wj then a cannot be tw, because that value would give ft = 0, which is a zero, not an infinity. Hence either a = «', and then /3 = &/ + &>; or a = &)' + &), and then /3 = ft)'. These are effectively one solution ; so that, if a + /3 = &), we have a, /3 = ft)', &>' + &)) and a, 6 = 0, &) ) ' jf a + /S = w + &>', then a cannot be CD f &)', because that value would give {$ = 0, which is a zero, not an infinity. Hence either a = &> and then ft = &)', or a = ft)' and then /3 = &). These again are effectively one solution ; so that, if a + /3 = &) + &>', we have a, £ = o), ft)' and a, 6 = 0, «o + ft)') 120.] OF THE SECOND CLASS 245 This combination can, by a change of fundamental parallelogram, be made the same as the former ; for, taking as new periods 2ft/ = 2a>'t 2fl = 2« + 2a>', which give a new fundamental parallelogram, we have a + j3 = H, and a, ft — &>', ft — ft/, that is, ft/, ft — 03' + 2<o' so that a, /3 = ft/, O + a/] and a, b = 0, being the same as the former with O instead of &>. Hence it is sufficient to retain the first solution alone : and therefore a = to', ft = CD' + co, a = 0, 6 = w. Hence, by § 116, 1., we have where F(z) is finite everywhere within the parallelogram. Again, $ (z + a/) has z = 0 and z = &> as its irreducible infinities, and it has 2 = 0)' and z = &> + &/ as its irreducible zeros, within the parallelogram of (f) (z} ; hence where ^ (2) is finite everywhere within the parallelogram. Thus a function which is finite everywhere within the parallelogram ; since it is doubly-periodic, it is finite everywhere in the plane and it is therefore a constant and equal to the value at any point. Taking - i&/ as the point (which is neither a zero nor an infinity) and remembering that </> is an odd function, we have * (*)*(* + «0 = - ft (*»')}' = p k being a constant used to represent the value of - {<£ (^o/)}"2. Also <j>(z + o)) = <f>(z + a + /3- 2&/) = c/>0 + a + /3)=-(£ (z), and therefore also <£ (&> — z) = <f) (z). The irreducible zeros of <j>' (z) were obtained in § 117, X. In the present example, those points are a>' + £ft>, &>' + ffc>, £&>, f &> ; so that, as there, we have £to'('W-{*(i)-4>(i*yito(*)-HW where K is a constant. But $ (f®) = 0 (2® - lft>) = (f) (-!«) = _(£ (1 a,) ; 246 DOUBLY-PERIODIC FUNCTIONS [120. and 0(fw + &/) = <£(2a> + 2w' -.!&>- a/) = <£(- 2 <o -to') = — </>(£&> + &>'); so that • ,. . 4 I - where J. is a new constant, evidently equal to {<£'(0)}2. Now, as we know the periods, the irreducible zeros and the irreducible infinities of the function </> (z), it is completely determinate save as to a constant factor. To determine this factor we need only know the value of <$>(z) for any particular finite value of z. Let the factor be determined by the condition then, since <£(^ft>)<£(^G> + ft/) = T by a preceding equation, we have and then ft' (*)}» - {f (0)}« [1 - {<£ (*)}2] [1 - fr {(/> (*)}'] Hence, since (/> (2) is an odd function, we have <£ (z) = sn (//,£). Evidently 2/xtu, 2/^ft)' = 4^T, 2^', where K and ^T' have the ordinary signifi cations. The simplest case arises when /A = 1. 121. Before proceeding to the deduction of the properties of even functions of z which are doubly-periodic, it is desirable to obtain the addition-theorem for <f>, that is, the expression of <p (y + z) in terms of functions of y alone and z alone. When <f> (y + z) is regarded as a function of z, which is necessarily of the second order, it is (§ 119, XV.) of the form where M and L are of degree in <£ (z) not higher than 2 and N is independent of z. Moreover y + z = a and y + z = ft are the irreducible simple infinities of <j) (y + z) ; so that L, as a function of z, may be expressed in the form and therefore Z±_^(iHL^^^)}l (z) - 121.] OF THE SECOND CLASS 247 where P, Q, R, S are independent of z but they may be functions of y. Now </> (a - y) = </> (w' - y) = - and <£ (/3 — y) = <j> (&>' + w — y} = so that the denominator of the expression for <f> (y + 2) is Since </> (z) is an odd function, <£' (#) is even ; hence ,A p - ~ */ and therefore $ (y + z) — $ (y — z) = - - Differentiating with regard to z and then making z = 0, we have so that, substituting for Q we have Interchanging y and z and noting that </> (t/ — z) = — (f) (z — y), we have md therefore d> C7y * Z} d>' (0} - W+*)1>< which is the addition-theorem required. Ex. If f(u) be a doubly-periodic function of the second order with infinities 61} i2, and 0(tt) a doubly -periodic function of the second order with infinities alt a2 such that, in the vicinity of «» (for i — 1, 2), we have ^ (M) = ,7~!r +Pi+& (u~ai) + ...... > c6 — u-j thon /M-/W = • i» W+* W-ft-ftl- the periods being the same for both functions. Verify the theorem when the functions are sn u and sn (u + v}. (Math. Trip. Part II., 1 891.) Prove also that, for the function $ (u), the coefficients p± and p2 are equal. (Burnside.) 122. The preceding discussion of uneven doubly-periodic functions having two simple irreducible infinities is a sufficient illustration of the 248 DOUBLY-PERIODIC FUNCTIONS [122. method of procedure. That, which now follows, relates to doubly- periodic functions with one irreducible infinity of the second degree ; and it will be used to deduce some of the leading properties of Weierstrass's er-function (of § 57) and of functions which arise from it. The definition of the <r-function is where fi = 2ma> + 2m'a)', the ratio of &>' : &> not being purely real, and the infinite product is extended over all terms that are given by assigning to m and to m' all positive and negative integral values from +00 to — oo , excepting only simultaneous zero values. It has been proved (and it is easy to verify quite independently) 'that, when cr(z) is regarded as the product of the primary factors the doubly-infinite product converges uniformly and unconditionally for all values of z in the finite part of the plane ; therefore the function which it represents can, in the vicinity of any point c in the plane, be expanded in a converging series of positive powers of z — c, but the series will only express the function in the domain of c. The series, however, can be continued over the whole plane. It is at once evident that a- (z) is not a doubly-periodic function, for it has no infinity in any finite part of the plane. It is also evident that a (z) is an odd function. For a change of sign in z in a primary factor only interchanges that factor with the one which has equal and opposite values of m and of m', so that the product of the two factors is unaltered. Hence the product of all the primary factors, being independent of the nature of the infinite limits, is an even function ; when z is associated as a factor, the function becomes uneven and it is a- (z). The first derivative, a' (z), is therefore an even function ; and it is not infinite for any point in the finite part of the plane. It will appear that, though a- (z) is not periodic, it is connected with functions that have 2o> and 2&>' for periods ; and therefore the plane will be divided up into parallelograms. When the whole plane is divided up, as in § 105, into parallelograms, the adjacent sides of which are vectorial repre sentations of 2w and 2&/, the function a-(z) has one, and only one, zero in each parallelogram; each such zero is simple, and their aggregate is given by z = £l. The parallelogram of reference can be chosen so that a zero of <r (z} does not lie upon its boundary ; and, except where explicit account is 122.] OF THE FIRST CLASS 249 taken of the alternative, we shall assume that the argument of &>' is greater than the argument of to, so that the real part* of w'/ia) is positive. 123. We now proceed to obtain other expressions for a- (z), and particu larly, in the knowledge that it can be represented by a converging series in the vicinity of any point, to obtain a useful expression in the form of a series, converging in the vicinity of the origin. Since er (z) is represented by an infinite product that converges uniformly and unconditionally for all finite values of z, its logarithm is equal to the sum of the logarithms of its factors, so that where the series on the right-hand side extends to the same combinations of m and m' as the infinite product for z, and, when it is regarded as a sum of z z^ ( z\ functions o + i 7^2 + ^°£ ( ^ ~ r> ) » ^ne sei>ies converges uniformly and uncon- __ -- \ 1 - , ditionally, except for points z = £l. This expression is valid for log a (z) over the whole plane. Now let these additive functions be expanded, as in § 82. In the imme diate vicinity of the origin, we have a series which converges uniformly and unconditionally in that vicinity. Then the double series in the expression for log a (z} becomes and as this new series converges uniformly and unconditionally for points in the vicinity of z = 0, we can, as in § 82, take it in the form oo ~r ( oo oo } 5" J 5* y O-n — 4 — \ ^-> <5r »• () r=3 r (-00 -oo J which will also, for such values of z, converge uniformly and unconditionally. In § 56, it was proved that each of the coefficients 00 00 2 s n-*-, — 00 - 00 for r = 3, 4,..., is finite, and has a value independent of the nature of the infinite limits in the summation. When we make the positive infinite limit for m numerically equal to the negative infinite limit for m, and likewise for This quantity is often denoted by ffi ( . - J . 250 WEIERSTRASS'S [123. ra', then each of these coefficients determined by an odd index r vanishes, and therefore it vanishes in general. We then have log a- 0) = log z - I* 22ft-4 - ^ 22ft-6 - ^ 22ft-8 a series which converges uniformly and unconditionally in the vicinity of the origin. The coefficients, which occur, involve «o and «', two independent constants. It is convenient to introduce two other magnitudes, g.2 and g3, denned by the equations #2= 6022ft-4, #3 = 140220-0, so that g2 and </3 are evidently independent of one another; then all the remaining coefficients are functions* of g.2 and g3. We thus have and therefore <r (z) = ze m where the series in the index, containing only even powers of z, converges uniformly and unconditionally in the vicinity of the origin. It is sufficiently evident that this expression for a- (z) is an effective representation only in the vicinity of the origin ; for points in the vicinity of any other zero of cr (z), say c, a similar expression in powers of z - c instead of in powers of z would be obtained. 124. From the first form of the expression for log cr (z), we have o-(z) z _«, _ where the quantity in the bracket on the right-hand side is to be regarded as an element of summation, being derived from the primary factor in the product-expression for cr (z\ We write £(z) = , ^ , so that %(z) is, by § 122, an odd function, a result also easily derived from the foregoing equation ; and so This expression for £ (z) is valid over the whole plane. Evidently £ (z) has simple infinities given by for all values of ra and of m between + oo and - oo , including simultaneous zeros. There is only one infinity in each parallelogram, and it is simple ; for the function is the logarithmic derivative of a (z\ which has no infinity and * See Quart. Journ., vol. xxii., pp. 4, 5. The magnitudes g2 and g3 are often called the invariants. 124.] ELLIPTIC FUNCTION 251 only one zero (a simple zero) in the parallelogram. Hence %(z) is not a doubly-periodic function. For points, which are in the immediate vicinity of the origin, we have but, as in the case of cr(z), this is an effective representation of %(z) only in the vicinity of the origin ; and a different expression would be used for points in the vicinity of any other infinity. We again introduce a new function g> (z) defined by the equation Because £ is an odd function, $ (z) is an even function ; and where the quantity in the bracket is to be regarded as an element of summation. This expression for $ (z) is valid over the whole plane. Evidently |p (z) has infinities, each of the second degree, given by z = fl, for all values of m and of m between -f oo and - oo , including simultaneous zeros ; and there is one, and only one, of these infinities in each parallelogram. One of these infinities is the origin; using the expression which represents log a- (z) in the immediate vicinity of the origin, we have = -2 + 20 9** + ^ 9*?+ • • • for points z in the immediate vicinity of the origin. A corresponding expression exists for g> (z) in the vicinity of any other infinity. 125. The importance of the function $ (z) is due to the following theorem : — The function $> (z) is doubly-periodic, the periods being 2<w and 2&/. Wo have -l where the doubly-infinite summation excludes simultaneous zero values, and the expression is valid over the whole plane. Hence + ^-n - Si 252 WEIERSTRASS'S [125. so that obtained by combining together the elements of the summation in g> (z + 2<w) and |p (,z). The two terms, not included in the summation, can be included, if we remove the numerical restriction as to non-admittance of simultaneous zero values for m and m'\ and then 2.) - f (,) = 2 _ where now the summation is for all values of m and of m' from + oo to — oo . Let q denote the infinite limit of m, and p that of m'. Then terms in the first fraction, for 0 = 2 (mm + m'w'}, are the same as terms in the second for £1 = 2 (m — l)w + 2m' w ; cancelling these, we have m'=p -fC = where q is infinite. But ?r)2 sin2 c ' and therefore »»' = p = oo 1 ^.2 2i - 2mV}2 W . sin 2o/ if/) be infinitely great compared with q. This condition may be assumed for the present purpose, because the value of g> (z) is independent of the nature of the infinite limits in the summation and is therefore unaffected by such a limitation. f - "" 1 ] f £+!?(9+1) -*$-* * l_* J ' l_ The fraction —, has a real part. In the exponent it is multiplied by q + 1. that is, by an infinite quantity ; so that the real part of the index of the exponential is infinite, either positive or negative. Thus either the first term is infinite and the second zero, or vice versa; in either case, r T i • sin \z + 2 (q + 1) twl ^— , is infinite, and therefore 2o) J {2 + 2(q + l)(o- 2m V}2 Similarly for the other sum. Hence = 0. In the same way it may be shewn that £>0 + 2a/)-£>0) = 0; therefore £> (z) is doubly-periodic in 2<o and 2a>'. 126.] ELLIPTIC FUNCTION 253 Now in any parallelogram whose adjacent sides are 2&> and 2&>', there is only one infinity and it is of multiplicity two; hence, by § 116, Prop. III., Cor. 3, 2o) and 2&>' determine a primitive parallelogram for $> (z). We shall assume the parallelogram of reference chosen so as to include the origin. 126. The function $ (z) is thus of the second order and the first class. Since its irreducible infinity is of the second degree, the only irreducible infinity of g>' (z} is of the third degree, being the origin ; and the function <§t ' (z) is odd. The zeros of jp' (z} are thus &>, ft/, and (&> + to') ; or, if we introduce a new quantity w" defined by the equation &>" = &) + &>', the zeros of <@! (z) are &>, &>', &>". We take #>(«) = e1} p(a>") = e2, p(m') = e3, %>(z) = p: and then, by § 119, Prop. XIV., Cor., we have where A is some constant. To determine the equation more exactly, we substitute the expression of jp in the vicinity of the origin. Then 80 that P' = -j+iQff* + When substitution is made, it is necessary to retain in the expansion all terms up to z° inclusive. We then have, for |p'2, the expression 4 2 4 and for A (^ - e-,) (p - e2) (p - e3), the expression A r1 3 9- 3 1L^6 + 20^+285r3+>" - (e, + e2 + e3)(^+—g2 + ...)+ (6le, + e2e3 + e&) (- + ...)- tf,«A | When we equate coefficients in these two expressions, we find e1 + e2 + e3 = 0, e& + e» therefore the differential equation satisfied by p is 254 PERIODICITY [126. Evidently £>" = 6§>2 - %gs, and so on ; and it is easy to verify that the 2wth derivative of g) is a rational integral algebraical function of <p of degree n + l and that the (2w+l)th derivative of fp is the product of g>' by a rational integral algebraical function of degree n. The differential equation can be otherwise obtained, by dependence on Cor. 2, Prop. V. of § 116. We have, by differentiation of %>', for points in the vicinity of the origin ; and also ^+!^2 + r4^2 + "-- Hence <@" and §>2 have the same irreducible infinities in the same degree and their fractional parts are essentially the same : they are homoperiodic and therefore they are equivalent to one another. It is easy to see that g>" — 6(jp2 is equal to a function which, being finite in the vicinity of the origin, is finite in the parallelogram of reference and therefore, as it is doubly -periodic, is finite over the whole plane. It therefore has a constant value, which can be obtained by taking the value at any point; the value of the function for z = 0 is — \g» and therefore g>"_6^ = -^2, so that |p"= 6g»2-|<72, the integration of which, with determination of the constant of integration, leads to the former equation. This form, involving the second derivative, is a convenient one by which to determine a few more terms of the expansion in the vicinity of the origin : and it is easy to shew that from which some theorems relating to the sums ^SH"2*1 can be deduced*. Ex. If cn be the coefficient of 22n~2 in the expansion of $ (z) in the vicinity of the origin, then c»=/o~ . iw.. ON 2 Crfin-r- (Weierstrass.) We have jp'2 = 4^>3 - g$ - g3 ; the function jjp' is odd and in the vicinity of the origin we have * See a paper by the author, Quart. Journ., vol. xxii, (1887), pp. 1 — 43, where other references are given and other applications of the general theorems are made. 126.] OF WEIERSTRASS'S FUNCTION 255 hence, representing by — (4|p3 — g$>— g^ that branch of the function which is negative for large real values, we have and therefore z = The upper limit is determined by the fact that when z = 0, g> = oo ; so that - r d® _ r d%> lp {4 (p - ej (p - e2) (p - e,)}*' This is, as it should be, an integral with a doubly-infinite series of values. We have, by Ex. 6 of § 104, r 0)j = ft) =| J<h , ft>3= ft) = J with the relation a)" = a) + co'. 127. We have seen that g> (z) is doubly-periodic, so that p(*+2») = $>(*), and therefore dg(5 + 2«) = dgW a^ dz hence integrating ?(^ + 2<») = %(z) + A. Now ^ is an odd function ; hence, taking z — — co which is not an infinity of £, we have ^ = 2^(&))=27; say, where r) denotes £ (&>) ; and therefore £(*-*• 2»)r- £•(*)« 89, which is a constant. Similarly %(z + 2&>') - ^ (^) - 2i/, where r;' = ^ (to') and is constant. Hence combining the results, we have % (z + 2w&) + 2?rc V) -£(z) = 2mri + Zmrj', where m and m are any integers. It is evident that 77 and rj' cannot be absorbed into £; so that £ is not a periodic function, a result confirmatory of the statement in § 124. 256 PSEUDO-PERIODIC [127. There is, however, a pseudo-periodicity of the function £ : its characteristic is the reproduction of the function with an added constant for an added period. This form is only one of several simple forms of pseudo-periodicity which will be considered in the next chapter. 128. But, though %(z) is not periodic, functions which are periodic can be constructed by its means. Thus, if 4>(z)=AS(z-a)+Bt(z-V) then * + 2w-(*) = 2A£(*-a so that, subject to the condition A+B+C+...=0, <j) (z) is a doubly-periodic function. Again, we know that, within the fundamental parallelograTH, f has a single irreducible infinity and that the infinity is simple; hence the irre ducible infinities of the function </> (V) are z = a, b, c, ..., and each is a simple infinity. The condition A + B + C + ...=0 is merely the condition of Prop. III., § 116, that the ' integral residue ' of the function is zero. Conversely, a doubly-periodic function with m assigned infinities can be expressed in terms of f and its derivatives. Let ax be an irreducible infinity of <£> of degree n, and suppose that the fractional part of <I> for expansion in the immediate vicinity of ax is A! i?i | ^ KI Then -if (*- 4).-... is not infinite for z = a^. Proceeding similarly for each of the irreducible infinities, we have a function r which is not infinite for any of the points z = alt a2, .... But because <f> (z) is doubly-periodic, we have and therefore the function 128.] FUNCTIONS 257 is doubly-periodic. Moreover, all the derivatives of any order of each of the functions £ are doubly-periodic; hence the foregoing function is doubly- periodic. The function has been shewn to be not infinite at the points a1} a2, ..., and therefore it has no infinities in the fundamental parallelogram ; con sequently, being doubly-periodic, it has no infinities in the plane and it is a constant, say G. Hence we have g, r=i r=i m with the condition 2 Ar = 0, which is satisfied because <E> (z) is doubly- periodic. This is the required expression * for <I> (z) in terms of the function % and its derivatives; it is evidently of especial importance when the indefinite integral of a doubly-periodic function is required. 129. Constants 77 and 77', connected with &> and «', have been introduced by the pseudo-periodicity of £(z)\ the relation, contained in the following proposition, is necessary and useful : — The constants 77, w', &>, &>' are connected by the relation the + or - sign being taken according as the real part of o>'fa)i is positive or negative. A fundamental parallelogram having an angular point at z0 is either of the form (i) in fig. 34, in which case 9t f-^] is \mj positive : or of the form (ii), in which case 9J ( — . ) \Ct)l/ is negative. Evidently a description of the paral lelogram A BCD in (i) will give for an integral the 'same result (but with an opposite sign) as a de scription of the parallelogram in (ii) for the same integral in the direction A BCD in that figure. We choose the fundamental parallelogram, so that it may contain the origin in the included area. The origin is the only infinity of £ which can be within the area : along the boundary £ is always finite. Now since * See Hermitet Ann. de Toulouse, t. ii, (1888), C, pp. 1—12. F. 20+2o)' Fig. 34 258 PSEUDO-PERIODICITY OF WEIERSTRASS's [129. the integral of £(z) round ABCD in (i), fig. 34, is (§ 116, Prop. II., Cor.) rD CB 2r)dz - fy'dz, J A J A the integrals being along the lines AD and AB respectively, that is, the integral is 4 (rju>' — rfw}. But as the origin is the only infinity within the parallelogram, the path of integration ABCD A can be deformed so as to be merely a small curve round ! the origin. In the vicinity of the origin, we have and therefore, as the integrals of all terms except the first vanish when taken round this curve, we have = 2-Trt. Hence 4 (rjw — TJ'O)) = 27ri, and therefore i](f> — rju> = \iri. This is the result as derived from (i), fig. 34, that is, when 91 [?-) is positive. \i/tU/ When (ii), fig. 34, is taken account of, the result is the same except that, when the circuit passes from z0 to z0 + 2&>, then to z0 + 2t» + 2o>', then to z0 + 2&>' and then to z0, it passes in the negative direction round the' parallelogram. The value of the integral along the path ABCDA is the same as before, viz., 4 (rjw — rj'a)) ; when the path is deformed into a small! rdz curve round the origin, the value of the integral is I — taken negatively, an J ** therefore it is — 2?ri : hence t](£) — rj (a = — \Tri. Combining the results, we have rjay' — f]w = ± ^Tri, / '\ according: as 9t ( — . ] is positive or negative. \0)lJ COROLLARY. If there be a change to any other fundamental parallelo gram, determined by 2H and 2O', where £1 = pa) + qa)', £1' = p'co + q'a)', p, q, p', q' being integers such that pq — p'q = ± 1, and if H, H' denote C(ft'), then H = pr} + qrj', H' = p'rj + q'f}' ; therefore HW - H'£l = ± \-iri, according as the real part of T^ is positive or negative. 130.] PRODUCT-FUNCTIONS 259 130. It has been seen that £ (z) is pseudo-periodic ; there is also a pseudo- periodicity for o- (z), but of a different kind. We have that is, 0-0 + and therefore a- (z + 2<w) = Ae^zcr (z), where A is a constant. To determine A, we make z = — &>, which is not a zero or an infinity of a (z) ; then, since a (z) is an odd function, we have so that o- 0 + 2&>) = - e*> <z+<0> a- (z). Hence o-(z + 4eo) = — e*> (z+3ft)) <r(z + 2&>) and similarly a (z + 2mey) = (— l)m Proceeding in the same way from we find a~(z + 2m V) = (- l)m' e*>' <w/z+m'2w') o- (z). Then a (z + 2ma> + 2m V) = (- l)m e2^ (ww+^»+»»»»V) Q- (^ + 2m/eo/) == / _ J \m+m' g sz (mij+m'V) +2'?m2<o+47]mmV+27)'m'2co' _. But lyct)' — r/o) = ± \iri, SO that g2mm'(r|a>'— rj'w) _ e±mm'iri _ /_ |\nj.m' and therefore 2m V) = (- l)w which is the law of change of a (z) for increase of z by integral multiples of the periods. Evidently <r(z) is not a periodic function, a result confirmatory of the statement in § 122. But there is a pseudo-periodicity the characteristic of which is the reproduction, for an added period, of the function with an exponential factor the index being linear in the variable. This is another of the forms of pseudo-periodicity which will be considered in the next chapter. 131. But though <r(z) is not periodic, we can by its means construct functions which are periodic in the pseudo-periods of a (z). By the result in the last section, we have <r (z — a. + 2ma> + 2m'&)') cr (z — a) + ,, <r(z-fi + 2mo> + 2m V) ~ <r(z ^~J3) & ' 17—2 260 DOUBLY-PERIODIC FUNCTIONS [131. and therefore, if <f> (z) denote a- (z — cti) a (z — 02) cr (z — &») then $ (z + 2ra&> + 2m V) = e2(m>} +m'*'> <2^-2^ so that $ (z) is doubly-periodic in 2« and 2&>' provided Now the zeros of <f>(z), regarded as a product of o--functions, are als a2,..., «„ and the points homologous with them ; and the infinities are Pi, /32, ... , ftn and i the points homologous with them. It may happen that the points a and ft j are not all in the parallelogram of reference ; if the irreducible points homologous with them be a1} ..., an and blt ... , bn, then Sar = ~S.br (mod. 2&>, 2co'), and the new points are the irreducible zeros and the irreducible infinities of <}>(z). This result, we know from Prop. III., § 116, must be satisfied. It is naturally assumed that no one of the points a is the same as, or is homologous with, any one of the points ft : the order of the doubly-periodic function would otherwise be diminished by 1. If any a be repeated, then that point is a repeated zero of <j>(z); similarly- if any ft be repeated, then that point is a repeated infinity of <£ (z). In every, case, the sum of the irreducible zeros must be congruent with the sum of the irreducible infinities in order that the above expression for <j)(z) may be doubly-periodic. Conversely, if a doubly-periodic function <£ (z) be required with m assignedJ irreducible zeros a and m assigned irreducible infinities b, which are subject t to the congruence 2a = 26 (mod. 2co, 2&>'), we first find points OL and ft homologous with a and with b respectively sucht that rru +U t « Then the function a- (z - Pi) a(z — ftm) has the same zeros and the same infinities as </> (z), and is homoperiodic withi it ; and therefore, by § 116, IV., o-(s-ai) o-^-otm) 9 \z> — •"• „(„ _ o \ *(*—ft V where A is a quantity independent of z. Ex. 1. Consider ft? (z). It has the origin for an infinity of the third degree and all thti remaining infinities are reducible to the origin ; and its three irreducible zeros are a, a/, a>" j Moreover, since <o"=a>' + a>, we have w + w' + w" congruent with but not equal to zerw We therefore choose other points so that the sum of the zeros may be actually the same, 131.] EXAMPLES 261 as the sum of the infinities, which is zero ; the simplest choice is to take <», &>', - «". Hence where A is a constant. To determine A, consider the expansions in the immediate vicinity of the origin ; then 2 o- ( - co) <r ( - to ) v (a)") ?"•" ...... S3 * ...... > sothat y^-g^^/rf^'tf. O- («) or (eo ) o- (a ) O"3 (2) Another method of arranging zeros, so that their sum is equal to that of the infinities, is to take — w, — «', co" ; and then we should find dy M =2 r W This result can, however, be deduced from the preceding form merely by changing the sign of z. Ex. 2. Consider the function . a- (u + v) a- (u — v) *«(«) where v is any quantity and A is independent of u. It is, qua function of u, doubly- periodic ; and it has u = 0 as an infinity of the second degree, all the infinities being homologous with the origin. Hence the function is homoperiodic with g> (u) and it has the same infinities as $> (u) : thus the two are equivalent, so that where B and C are independent of u. The left-hand side vanishes if n—v; hence (v), and therefore where A' is a new quantity independent of u. To determine .4' we consider the expansions in the vicinity of u = 0 ; we have A.'<r(v)<r(-v) sothat and therefore cr- = o-2 (%) o-2 (v) a formula of very great importance. Ex. 3. Taking logarithmic derivatives with regard to u of the two sides of the last equation, we have and, similarly, taking them with regard to D, we have whence 262 EXAMPLES [131. giving the special value of the left-hand side as (§ 128) a doubly-periodic function. It is also the addition-theorem, so far as there is an addition-theorem, for the ^-function. Ex. 4. We can, by differentiation, at once deduce the addition-theorem for g) (u + v). Evidently which is only one of many forms : one of the most useful is which can be deduced from the preceding form. The result can be used to modify the expression for a general doubly-periodic function * (z) obtained in § 128. We have Each derivative of f can be expressed either as an integral algebraical function of $ (z - a,.) or as the product of jjp' (z — ar) by such a function ; and by the use of the addition-theorem these can be expressed in the form L > where L, M, N are rational integral algebraical functions of $(z). Hence the function can be expressed in the same form, the simplest case being when all its infinities are simple, and then 4. (z) = C+ 2 Ar{(e-ar) (*)-§» (Or) with the condition 2 Ar = 0. r=l Ex. 5. The function $ (z) — e1 is an even function, doubly-periodic in 2« and 2o> and having 2 = 0 for an infinity of the second degree ; it has only a single infinity of the second degi'ee in a fundamental parallelogram. Again, z = &> is a zero of the function ; and, since ^X («) = 0 but $>" (o>) is not zero, it is a double zero of $ (z)-el. All the zeros are therefore reducible to 2 = o> ; and the function has only a single zero of the second degree in a fundamental parallelogram. Taking then the parallelogram of reference so as to include the points 0 = 0 and 0=o>, we have where Q (z) has no zero and no infinity for points within the parallelogram. Again, for g> (z + o>) - e± , the irreducible zero of the second degree within the parallelo- 131.] OF DOUBLY-PERIODIC FUNCTIONS 263 gram is given by S + <B = O>, that is, it is 0 = 0; and the irreducible infinity of the second degree within the parallelogram is given by z + a = 0, that is, it is z = v. Hence we have where Ql (z) has no zero and no infinity for points within the parallelogram. Hence {£> (z} - ej {%> (z + «) - ej m Q (z) Q1 (z\ that is, it is a function which has no zero and no infinity for points within the parallelogram of reference. Being doubly-periodic, it therefore has no zero and no infinity anywhere in the plane ; it consequently is a constant, which is the value for any point. Taking the special value s = a>, we have jp(m') = es, and (jf>(a>' + a>) = e2 ; and therefore {#> (*) ~ e,} (V (* + «)- e,} = (e3 - *i) (e, - *i). Similarly {#> (z) - e2} {#> (z + »") - e2} = (ex - e2) (es - e2), and {#> (2) - ^ {p (z + a)') - e3} = (e2 - e3} (^ - <?3). It is possible to derive at once from these equations the values of the ^-function for the quarter-periods. Note. In the preceding chapter some theorems were given which indicated that functions, which are doubly-periodic in the same periods, can be expressed in terms of one another : in particular cases, care has occasionally to be exercised to be certain that the periods of the functions are the same, especially when transformations of the variables are effected. For instance, since g) (z) has the origin for an infinity and sn u has it for a zero, it is natural to express the one in terms of the other. Now $ (z) is an even function, and sn u is an odd function ; hence the relation to be obtained will be expected to be one between ®(z) and sn2w. But one of the periods of sn2 u is only one-half of the correspond ing period of sn u ; and so the period-parallelogram is changed. The actual relation* is (P (z) - <?3 = (<?!- e3) sn-2«, where u = (el -esf z and F = (<?2 -e^)l(el -e:i). Again, with the ordinary notation of Jacobian elliptic functions, the periods of sn z are 4 A" and 2iA", those of dn z are 2 A and 4i K', and those of en z are 4 A' and 2A'+ 2iA''. The squares of these three functions are homoperiodic in 2K and ZiK' ; they are each of the second order, and they have the same infinities. Hence sn2 z, en2 z, dn2 z are equivalent to one another (§ 116, V.). But such cases belong to the detailed development of the theory of particular classes of functions, rather than to what are merely illustrations of the general propositions. 132. As a last illustration giving properties of the functions just considered, the derivatives of an elliptic function with regard to the periods will be obtained. Let (/> (z) be any function, doubly-periodic in 2o> and 2&/ so that </> (z + 2m&> + 2m V) = </> (z), the coefficients in <f> implicitly involve &> and CD'. Let <f>1} <£2, and </>' respec tively denote 90/3t», 9<£/9o/, 9^/9^ ; then ^ (z + 2771&) + 2m' to') + %m<j>' (z + 2mo> + 2wV) = fa (z), fa (z + 2m&) + 2??iV) -1- 2m'<£' (z + 2rao> + 2wV) = fa (z), $ (z + 2m&) + 2m V) = <f> (z). " Halphen, Fonctions Elliptiques, t. i, pp. 23 — 25. 264 PERIOD-DERIVATIVES [132. Multiplying by &>, ro', z respectively and adding, we have &></>! (z + 2mo> + 2m V) + o>'</>2 (z + 2m<« + 2mV) + (z + 2mw + 2wV) </>' 0 + 2mo) = 0)0! (Z) + 0)'(f).2 (Z) + Z<f>' (Z). Hence, if f(z} = mfa (z) + 6/02 (z) + z$ (z), then f(z) is a function doubly -periodic in the periods of (f>. Again, multiplying by rj, 77', %(z), adding, and remembering that £0 + 2mm + 2raV) = £($) we have 77$! (z + 2mw + 2m V) + 7/<£2 0 + 2m«o + 2m V) + £(z + 2mm + 2m' ay') <f>' (z + 2mw + 2m'a>') -ik<fi)+J+,(*) + f(*Wto Hence, if g (z) = yfa (z) + q'fa (z) + £ (z) $' (z), then g(z) is a function doubly-periodic in the periods of <f>. In what precedes, the function <f>(z) is any function, doubly-periodic in 2o>, 2&)' ; one simple and useful case occurs when 0 (z) is taken to be the function z. Now and fW-J-^ hence, in the vicinity of the origin, we have 9P >d@ d& 2 (o ^- + 03 5*7 + jp-«- as — + even integral powers of z1 d(o d(o dz z- = -2^>, since both functions are doubly-periodic and the terms independent of z vanish for both functions. It is easy to see that this equation merely expresses the fact that <p, which is equal to l is homogeneous of degree — 2 in z, &>, to'. Similarly 9|J> / ty ^d%> 22 77 2+*) ~r-/ + b (-2r) y- = - ~i + j^ 9-i + even integral powers of z. But, in the vicinity of the origin, ,5-7 = — + YQ^ -I- even integral powers of 2, 132.] OF WEIEHSTRASS'S FUNCTION 265 so that 9P / d@ »/ \ dP 1 32lP • x r 17 3^- + V g :S + f<«) |^t i gji ™i** even mtegral powers of z. The function on the left-hand side is doubly-periodic : it has no infinity at the origin and therefore none in the fundamental parallelogram ; it there fore has no infinities in the plane. It is thus constant and equal to its value anywhere, say at the origin. This value is ^gz> and therefore T/w's equation, when combined with , + eo' ; + z = - ty, dco da> oz j gives the value of ~- and -^ , . J dm 9&) The equations are identically satisfied. Equating the coefficients of z2 in the expansions, which are valid in the vicinity of the origin, we have and equating the coefficients of ^ in the same expansions, we have Hence for any function u, which involves w and &/ and therefore implicitly involves g2 and ^r3, we have du ,du w 5- + w — , = aw a&) 9w . , 3w 17 a- +T;' — = - 9&) 9&) Since ^) is such a function, we have f : *' being ^/te equations which determine the derivatives of $ with regard to the invariants g., and g.^. 266 EVEN [132. The latter equation, integrated twice, leads to 9V da- 2 80- 1 a differential equation satisfied by <r(z)*. 133. The foregoing investigations give some of the properties of doubly- periodic functions of the second order, whether they be uneven and have two simple irreducible infinities, or even and have one double irreducible infinity. If a function U of the second order have a repeated infinity at z = y, then it is determined by an equation of the form or, taking U - £ (X + fi + v) = Q, the equation is Q'» = 4a2 [(Q - e,) (Q - e,) (Q - $,)]*, where ^ + e2 + e3 = 0. Taking account of the infinities, we have Q=@(az- ay) ; and therefore U-±(\ + /Ji. + v) = %> (az - ay) . . 1 (tp'(az) + cp (ay)}'2 = -Q (az) - <o (ay) + — — a x ' o x " by Ex. 4, p. 262. The right-hand side cannot be an odd function; hence an odd function of the second order cannot have a repeated infinity. Similarly, by taking reciprocals of the functions, it follows that an odd function of the second order cannot have a repeated zero. It thus appears that the investigations in §§ 120, 121 are sufficient for the included range of properties of odd functions. We now proceed to obtain the general equations of even functions. Every such function can (by § 118, XIII., Cor. I.) be expressed in the form |a#> (z) +b}+ {c#> (z) + d], and its equations could thence be deduced from those of p(z)\ but, partly for uniformity, we shall adopt the same method as in § 120 for odd functions. And, as already stated (p. 251), the separate class of functions of the second order that are neither even nor odd, will not be discussed. 134. Let, then, <j>(z) denote an even doubly-periodic function of the second order (it may be either of the first class or of the second class) and let 2&), 2<w' be its periods ; and denote 2&) + 2ft)' by 2o>". Then since the function is even ; and since <£ (ft) + Z) = <f> (— &) — z) = <f> (2&) — &) — z) = (j) (CD — *), * For this and other deductions from these equations, see Frobenius und Stickelberger, Crelle, t. xcii, (1882), pp. 311—327; Halphen, Traite des feme t ions elliptiqucs, t. i, (1886), chap. ix. ; and a memoir by the author, quoted on p. 254, note. 134.] DOUBLY-PERIODIC FUNCTIONS it follows that <£ (&> + z)— and, similarly, $ (&>' + z) and 0 (to" + z) are even functions. Now </> (w + a), an even function, has two irreducible infinities, and is periodic in 2&>, 2&/ ; also <£ (z), an even function, has two irreducible infinities and is periodic in 2&>, 2&/. There is therefore a relation between 0 (z) and </> (w +z), which, by § 118, Prop. XIII., Cor. I., is of the first degree in <£ (z) and of the first degree in <j) (&> + z) ; thus it must be included in B<f> (z) <j>(<o + z)-C<l> (z) -C'<t>(a> + z)+A = 0. But <£ (z) is periodic in 2<w ; hence, on writing z + <w for z in the equation, it becomes B<f>(a>+z)<j>(z)-C<f>(co+z)-C'<l>(z) + A=0; thus tf=C". If B be zero, then (7 may not be zero, for the relation cannot become evanescent : it is of the form A' .............................. (1). If B be not zero, then the relation is Treating <f> (w + z) in the same way, we find that the relation between it and (f) (z) is F(j> (z) (f> (ay' + z)-D(j> (z} -D(j>(a>' + z) + E = 0, so that, if F be zero, the relation is of the form £(*) + 0(a>' + *) = J0' ........................... (I)', and, if F be not zero, the relation is of the form Four cases thus arise, viz., the coexistence of (1) with (1)', of (1) with (2)', of (2) with (1)', and of (2) with (2)'. These will be taken in order. I. : the coexistence of (1) with (1)'. From (1) we have <j> (a> + z} + (j> (&>" + z) = A', so that </> (z) + <f) (w + z) + (f) (w + z) + 0 (<w" + z) = 2A'. Similarly, from (1)', so that A = E', arid then (f)((0 + z)=(j)((o'+ Z\ whence <w ~ &>' is a period, contrary to the initial hypothesis that 2&> and 2&>' determine a fundamental parallelogram. Hence equations (1) and (1)' cannot coexist. 268 EVEN [134. II. : the coexistence of (1) with (2)'. From (1) we have <£(«" + z} = A' - <£(&>' + z) on substitution from (2)'. From (2)' we have cb (co + z) = -5*1) -- ( — =r F<f) (CD + z) - D _ (A'D -E)-D<j> (z) = A'F - D - F(f> (z) ' on substitution from (1). The two values of <£ (&>" + z) must be the same, whence A'F-D = D, which relation establishes the periodicity of </> (z) in 2ft)", when it is considered as given by either of the two expressions which have been obtained. We thus have A'F=W- and then, by (1), we have <f>(z)-j+<l>( and, by (2)', we have If a new even function be introduced, doubly-periodic in the same periods having the same infinities and defined by the equation & 0) = </> 0) - J > the equations satisfied by fa (z) are fa(a> + z) + fa(z) = 0 } fa (&)' + z) fa (z) = constant] ' To the detailed properties of such functions we shall return later ; meanwhile it may be noticed that these equations are, in form, the same as those satisfied by an odd function of the second order. III. : the coexistence of (2) with (1)'. This case is similar to II., with the result that, if an even function be introduced, doubly-periodic in the same periods having the same infinities and defined by the equation C fa (Z) = <£ (2) - -g , the equations satisfied by fa (z) are fa (&>' + z) + fa (z) = 0 } fa (&) + z) fa (z) = constant] ' It is, in fact, merely the previous case with the periods interchanged. 134.] DOUBLY-PERIODIC FUNCTIONS 269 IV. : the coexistence of (2) with (2)'. From (2) we have _ (CD - AF) <ft (z) - (GE - AD) ~ (BD - CF) <J> (z) - (BE - CD) ' on substitution from (2)'. Similarly from (2)', after substitution from (2), we have ~ The two values must be the same ; hence CD-AF=-(GD-BE\ which indeed is the condition that each of the expressions for <ft (&>" + z) should give a function periodic in 2&>". Thus One case may be at once considered and removed, viz. if C and D vanish together. Then since, by the hypothesis of the existence of (2) and of (2)', neither B nor F vanishes, we have A__E B~ F' so that u + , = and then the relations are <£ (&> + z) + <f) (&>' + z) = 0, or, what is the same thing, <ft (Y) + <ft (&>" + z) — 0 ] and </> (z) </>(&> + z) = constant j ' This case is substantially the same as that of II. and III., arising merely from a modification (§ 109) of the fundamental parallelogram, into one whose sides are determined by 2&> and 2&>". Hence we may have (2) coexistent with (2)' provided AF + BE=WD; C and D do not both vanish, and neither B nor F vanishes. IV. (1). Let neither C nor D vanish ; and for brevity write <f>((o + z)=<l>1, (f> (w" + z) = <£o, </> (&)' + z) = $3, (f) (z) = 0. Then the equations in IV. are Now a doubly-periodic function, with given zeros and given infinities, is determinate save as to an arbitrary constant factor. We therefore introduce an arbitrary factor X, so that <£=Xi/r, G D and then taking = CI' = Ca' 270 EVEN [134. £ we have (^ - Cl) (fa - Cj) = d2- -^ , ET The arbitrary quantity A, is at our disposal : we introduce a new quantity c2, defined by the equation A Tt-. o — Ci (C2 + €3) C2C3 , and therefore at our disposal. But since AF + BE=2CD, A E .CD we have ^ + ^ = 2 ^ ^ = 2Clc3j ri and therefore ^--2 = c3 (Cj + c2) - 0^2 . Hence the foregoing equations are - d) = (Cj - C2) (d - C3), - C3) = (C3 - d) (C3 - C2). The equation for ^>2, that is <f>((o" + z), is _Lcf)-M where L= CD - BE = AF - CD, M=AD-CE, N=CF-BD, so that ^ + 5M" = 2CL. As before, one particular case may be considered and removed. If N be zero, so that C_D_ B~F~a AE CD ,. say, and B+F=RF= ' then we find $ + ^>2 = ^>i + <£3 = 2«, or taking a function ^ = 0 — a, the equation becomes % (^ + % C^" + ^ = 0. The other equations then become and therefore they are similar to those in Cases II. and III. If N be not zero, then it is easy to shew that N=BF\(c1-c3)> M = BF\3 (d - c3) (c.,C! + c,c3 - dc3) ; 134.] DOUBLY-PERIODIC FUNCTIONS 271 and then the equation connecting 0 and 02 changes to s - Ca) = (Ca - Cx) (Ca - Cs) which, with (^ — d) ("^i — d) = (d — c2) (d — c3) ( r — ^3/ \ i 3 — ^3/ == V^3 ^" ^3 ^2' are relations between ty, ^rl} -^2, ty.3, where the quantity c2 is at our disposal. IV. (2). These equations have been obtained on the supposition that neither G nor D is zero. If either vanish, let it be C: then D docs not vanish ; and the equations can be expressed in the form E D\ J E\ E(D*-EF) We therefore obtain the following theorem : If (f> be an even function doubly-periodic in 2&> and 2&>' and of the second order, and if all functions equivalent to <J> in the form R<f> + 8 (where R and S are constants) be regarded as the same as 0, then either the function satisfies the system of equations 00) 0O" where H is a constant ; or it satisfies the system of equations {0 0) - d} {0 (ft) +Z)- d] = (Ci - C2) (d - C3) {00)-C3}{0(>/ +^)-C3}=(Cs-C1)(Cs-Ca) {0 0) - C2} (0 ((,)" + Z)- C2} = (C2 ~ d) (C2 - Cs) where of the three constants clt c2, cs one can be arbitrarily assigned. We shall now very briefly consider these in turn. 135. So far as concerns the former class of equations satisfied by an even doubly-periodic function, viz., we proceed initially as in (§ 120) the case of an odd function. We have the further equations 00) = 0(-4 0 (ft) + Z) — 0 (ft) — Z), 0 (a/ + Z) = 0 (ft)' — Z). * The systems obtained by the interchange of w, w', w" among one another in the equations are not substantially distinct from the form adopted for the system I. ; the apparent difference can be removed by an appropriate corresponding interchange of the periods. 272 EVEN DOUBLY-PERIODIC FUNCTIONS [135. Taking z = — ^w, the first gives so that ^&> is either a zero or an infinity. If \<£> be a zero, then (f> (f to) = $ (<« + ^ft)) = — <f> (^») by the first equation = 0, so that ^&> and f&> are zeros. And then, by the second equation, &)' + ^<w, &)' 4- f a) are infinities. If \w be an infinity, then in the same way |w is also an infinity ; and then a)' + \w, &>' + f &) are zeros. Since these amount merely to interchanging zeros and infinities, which is the same functionally as taking the reciprocal of the function, we may choose either arrangement. We shall take that which gives ^0), f &> as the zeros ; and &>' 4- ^&>, &/ + f &> as the infinities. The function <j> is evidently of the second class, in that it has two distinct simple irreducible infinities. Because &>' + |&), &>' + f &> are the irreducible infinities of </> (z), the four zeros of $' (z} are, by § 117, the irreducible points homologous with &>", &)" + &>, &>" + a)', a)" + &)", that is, the irreducible zeros of (f)' (z) are 0, &>, &>', &>". Moreover by the first of the equations of the system ; hence the relation between (f> ( and ((>' (z) is #* (z} = A{<t>(z)-$ (())} {</> (z) - (/> («)} |0 (*) - (/> (ft)')} {(/> (*) - </> («")} = A [p (0) - p (z)}{p (ft)') - ^ (*)}. Since the origin is neither a zero nor an infinity of <£ (^), let so that </>j (0) is unity and 0/ (0) is zero ; then ^(*)«X»{l-^(*)}{^-^(f)) the differential equation determining fa (z). The character of the function depends upon the value of p and the constant of integration. The function may be compared with en u, by taking 2ft), 2&/ = 4>K, 2K + 2iK' ; and with — *— , by taking 2ft), 2ft)' = 2K, MK', dn u which (§ 131, note) are the periods of these (even) Jacobian elliptic functions. We may deal even more briefly with the even, function characterised by the second class of equations in § 134. One of the quantities c1} c2, c3 being at our disposal, we choose it so that Ci + c2 + c3 = 0 ; and then the analogy with the equations of Weierstrass's ^-function is complete (see § 133). CHAPTER XII. PSEUDO-PERIODIC FUNCTIONS. 136. MOST of the functions in the last two Chapters are of the type called doubly-periodic, that is, they are reproduced when their arguments are increased by integral multiples of two distinct periods. But, in §§ 127, 130, functions of only a pseudo-periodic type have arisen : thus the ^-function satisfies the equation m2&> + m'2&>') = £(» + m2i) + m'2v', ,nd the cr-function the equation m' i (mr,+m'r,') (z+wuo+m'oi1) These are instances of the most important classes: and the distinction between the two can be made even less by considering the function e^(z} — ^(z), when we have £ (z + ra2&> + m'2&>') = e-mr> e"™'*' % (z). In the case of the ^-function an increase of the argument by a period leads to the reproduction of the function multiplied by an exponential factor that is constant, and in the case of the <r-function a similar change of the argument leads to the reproduction of the function multiplied by an exponential factor having its index of the form az + b. Hence, when an argument is subject to periodic increase, there are three simple classes of functions of that argument. First, if a function f(z) satisfy the equations /(* + 2fi>) =/(*), /(* + 2«') =/(*), it is strictly periodic : it is sometimes called a doubly-periodic function of the first kind. The general properties of such functions have already been considered. Secondly, if a function F(z) satisfy the equations F (z + 2&>) = pF (z), F (z + 2&/) - pfF (z), F- 18 274 THREE KINDS [136. where /u, and fjf are constants, it is pseudo-periodic : it is called a doubly- periodic function of the second kind. The first derivative of the logarithm of such a function is a doubly-periodic function of the first kind. Thirdly, if a function <f> (z) satisfy the equations <j>(z + 2o)) = eaz+b <j> (z\ <f>(z + 2ft)') = ea'z+v (j> (z), where a, b, a', b' are constants, it is pseudo-periodic : it is called a doubly- periodic function of the third kind. The second derivative of the logarithm of such a function is a doubly-periodic function of the first kind. The equations of definition for functions of the third kind can be modified. We have . <f> (Z + 2ft) + 2ft)') = e«(2+2<o')+6+a'z+6' <£ (z) — ga' (2+2o>) +b'+az+b J, fz\ whence a'oo — am' = — nnri, where ra is an integer. Let a new function E (z) be introduced, defined by the equation £(«)«*"+*• t(*)i then X and /A can be chosen so that E (z} satisfies the equations E(z + 2a)) = E (z\ E(z+ 2ft)') = eAz+£ E (z\ From the last equations, we have E (z + 2&) + 2ft)') = eA(*+**+B E ^ = eAz+s E (z), so that 2Aa) is an integral multiple of 2?™'. Also we have E(z + 2o>) = e*(*-*»'+^+a») <j>(z + 2o>) so that 4X&) + a = 0, and 4A,ftr + 2/A&) +6 = 0 (mod. Similarly, E (z + 2ft)') = e^+wj'+^+w, 0 ^ + 2ft)') so that 4Xo)' + a = A, and 4W2 + 2/^co' + 6' = B (mod. 27ri). From the two equations, which involve X and not //,, we have Aco = a'o) — aw' agreeing with the result with 2 A co is an integral multiple of Ziri. And from the two equations, which involve /j,, we have, on the elimination of /j, and on substitution for X and A, b'co — 6ft)' — a&)' (ft)' — &)) = 5ft) (mod. 2-Tn'). 136.] OF DOUBLY-PERIODIC FUNCTIONS 275 If A be zero, then E(z) is a doubly-periodic function of the first kind when eB is unity, and it is a doubly-periodic function of the second kind when eB is not unity. Hence A, and therefore m, may be assumed to be different from zero for functions of the third kind. Take a new function 3?z such that mm then <l> (z) satisfies the equations 4) (z + 2&)) = <I> (z\ <&(z + 2o)') = e w 3>(z) * / \ /' \ / \ /t which will be taken as the canonical equations defining a doubly -periodic function of the third kind. Ex. Obtain the values of X, p, A, B for the Weierstrassian function ir(z). We proceed to obtain some properties of these two classes of functions which, for brevity, will be called secondary-periodic functions and tertiary- periodic functions respectively. Doubly-Periodic Functions of the Second Kind. For the secondary-periodic functions the chief sources of information are Hermite, Comptes Rendus, t. liii, (1861), pp. 214—228, ib., t. Iv, (1862), pp. 11—18, 85 — 91 ; Sur quelques applications des fonctions elliptiques, §§ I — in, separate reprint (1885) from Comptes Rendus; "Note sur la theorie des fonctions ellip tiques" in Lacroix, vol. ii, (6th edition, 1885), pp. 484—491; Cours d' Analyse, (4me ed.), pp. 227—234. Mittag-Leffler, Comptes Rendus, t. xc, (1880), pp. 177 — 180. .Frobenius, Crelle, t. xciii, (1882), pp. 53 — 68. Brioschi, Comptes Rendus, t. xcii, (1881), pp. 325—328. Halphen, Traite' des fonctions elliptiques, t. i, pp. 225 — 238, 411 — 426, 438 442, 463. 137. In the case of the periodic functions of the first kind it was proved that they can be expressed by means of functions of the second order in the same period — these being the simplest of such functions. It will now be proved that a similar result holds for secondary- periodic functions, defined by the equations Take a function Q (z} = a (z) a- (a) then we have G(z+2a>) = <r(* + g a (a) a (z + 2w) arid G(z+ 2&/) = e'V«+2W Q. (^). The quantities a and X being unrestricted, we choose them so that „ _ g2rja+2A<o ' __ g2T)'a+2A(o' • and then G (z), a known function, satisfies the same equation as F (z). 18—2 276 PSEUDO-PERIODIC FUNCTIONS [137. Let u denote a quantity independent of z, and consider the function f(Z) = F(z)G(u-z}. We have f(z + 2o>) = F(z + 2o>) G (u- z - 2w) =/(*) ; and similarly f(z + 2<o') =f(z), so that/(X) is a doubly-periodic function of the first kind with 2« and 2o>' for its periods. The sum of the residues of f(z) is therefore zero. To express this sum, we must obtain the fractional part of the function for expansion in the vicinity of each of the (accidental) singularities of f(z), that lie within the parallelogram of periods. The singularities of/ (2) are those of G (u — z) and those of F(z). Choosing the parallelogram of reference so that it may contain u, we have z = u as the only singularity of G (u — z) and it is of the first order, so that, since $(£) — =+ positive integral powers of f in the vicinity of £= 0, we have, in the vicinity of u, f(z) = {F (u) + positive integral powers of u — z} \ — -4- positive powers I = -- — + positive integral powers of z — u ; hence the residue of/(Y) for u is —F(u}. Let z = c be a pole of F (z) in the parallelogram of order n + 1 ; and, in the vicinity of c, let (?! _ cf / 1 \ „ dn ( 1 \ F(z) = ^—c +G^Z (jr^J + • • • + C'n+i fan (zITc) + P°sltlve integral powers. Then in that vicinity and therefore the coefficient of - in the expansion of f(z) for points in the Z ~~ 0 vicinity of c is which is therefore the residue off(z) for c. This being the form of the residue of f(z) for each of the poles of F (z), then, since the sum of the residues is zero, we have 137.] OF THE SECOND KIND 277 or, changing the variable, ,. ..n+l n - where the summation extends over all the poles of F(z) within that parallelo gram of periods in which z lies. This result is due to Hermite. 138. It has been assumed that a and \, parameters in 0, are determinate, an assumption that requires /j, and ^ to be general constants : their values are given by yd 4- &>X = | log fjb, r)'a + &/X — \ log //, and, therefore, since ijca' — rfca = ± ^ITT, we have + ITTCL = w' log /JL — co log //) + iir\ = — V) log /i + 77 log /z'j ' Now X may vanish without rendering G (z) a null function. If a vanish (or, what is the same thing, be an integral combination of the periods), then G (z) is an exponential function multiplied by an infinite constant when X does not vanish, and it ceases to be a function when X does vanish. These cases must be taken separately. First, let a and X vanish* ; then both //, and /// are unity, the function F is doubly-periodic of the first kind ; but the expression for j^is not determinate, owing to the form of G. To render it determinate, consider X as zero and a as infinitesimal, to be made zero ultimately. Then „,, o-(z) + aa'(z) + ... .^ (*(z) = - - ~ — — (1 + positive integral powers of a) = - + £ (z) + positive powers of a. a Since a is infinitesimal, /JL and /j,' are very nearly unity. When the function F is given, the coefficients C1} <72, ... may be affected by a, so that for any one we have Ck — bk + ayk + higher powers of a, where yh is finite ; and bk is the actual value for the function which is strictly of the first kind, so that Sk-O, the summation being extended over the poles of the function. Then retaining only a"1 and a°, we have This case is discussed by Hermite (I.e., p. 275). 278 MITTAG-LEFFLER'S THEOREM [138. where C0, equal to £71, is a constant and the term in - vanishes. This expres- CL sion, with the condition S^ = 0, is the value of F (u) or, changing the variables, we have with the condition S&i = 0, a result agreeing with the one formerly (§ 128) obtained. When F is not given, but only its infinities are assigned arbitrarily, then SO = 0 because F is to be a doubly-periodic function of the first kind ; the term - "£C vanishes, and we have the same expression for F(z) as before. Secondly, let a vanish* but not \, so that ^ and // have the forms We take a function g (z) = then g(z- 2o>) = ^ e^ £ (z - 2eo ) and g(z-2a>') = p'-1 {g (z} - 2?/ e^} . Introducing a new function H (z) defined by the equation we have H (z + 2t») = H (z) - 2ijeA <«-*» F (z), and H (z + 2o>') = H (z) - 27?V<M-*> F(z). Consider a parallelogram of periods which contains the point u ; then, if © be the sum of the residues of H (z) for poles in this parallelogram, we have the integral being taken positively round the parallelogram. But, by § 116, Prop. II. Cor., this integral is f e-*(p+*-«) F (p + 2tot) dt - 0/77 f e-^+ Jo Jo where p is the corner of the parallelogram and each integral is taken for real values of t from 0 to 1. Each of the integrals is a constant, so far as concerns u ; and therefore we may take ® = -Ae^u, the quantity inside the above bracket being denoted by —\i-rrA. The residue of H (z) for z = u, arising from the simple pole of g (u — z), is -F(u) as in§ 137. If z = c be an accidental singularity of F (z) of order n+1, so that, in the vicinity of z = c, F(.) = C, + 0. A- + . . . + BU i- + P (, - c), This is discussed by Mittag-Leffler, (I.e., p. 275). 138.] ON SECONDARY FUNCTIONS 279 then the residue of H (z) for z = c is d dn and similarly for all the other accidental singularities of F (z}. Hence F(z) = A** + where the summation extends over all the accidental singularities of F (z) in a parallelogram of periods which contains z, and y (z) is the function exz%(z}. This result is due to Mittag-Leffler. Since /* = e2*" and g (z - c + 2&>) = fig (z - c) + we have and therefore 2 (Gl + C.2\ 4- . . . + Gn+l\n) e~^ = 0, the summation extending over all the accidental singularities of F(z). The same equation can be derived through ^F(z) = F(z + 2&>'). Again 2(7: is the sum of the residues in a parallelogram of periods, and therefore the integral being taken positively round it. If p be one corner, the integral n F (p + 2co't) dt, Jo IS /•i n o , each integral being for real variables of t. Hermite's special form can be derived from Mittag-Leffler's by making \ vanish. Note. Both Hermite and Mittag-Leffler, in their investigations, have used the notation of the Jacobian theory of elliptic functions, instead of dealing with general periodic functions. The forms of their results are as follows, using as far as possible the notation of the preceding articles. I. When the function is denned by the equations F (z + 2K) = ^F (z), F(z+ 2iK') = ^F (z), then F(z) = 280 INFINITIES AND ZEROS [138. (the symbol H denoting the Jacobian .ff-function), and the constants <w and X are determined by the equations II. If both X and to be zero, so that F(z) is a doubly-periodic function of the first kind, then with the condition 5$i = 0. III. If W be zero, but not X, then ... where g (z} = --& V, the constants being subject to the condition 2 (G, + C,\ + . . . + Gn+1 X")e-Ac = 0, and the summations extending to all the accidental singularities of F(z) in a parallelogram of periods containing the variable z. 139. Reverting now to the function F(z) we have G (z), defined as a (z) a (a) when a and X are properly determined, satisfying the equations G(z + 2a>) = ftG (z), £0 + 2&/) = yu/£0). Hence H (z) = F(z)/G (z) is a doubly-periodic function of the first kind ; and therefore the number of its irreducible zeros is equal to the number of its irreducible infinities, and their sums (proper account being taken of multipli city) are congruent to one another with moduli 2« and 2&>'. Let Ci, c2,..., cm be the set of infinities of F (z) in the parallelogram of periods containing the point z ; and let y:, . . . , 7^ be the set of zeros of F (z) in the same parallelogram, an infinity of order n or a zero of order n occurring n times in the respective sets. The only zero of 0 (z) in the parallelogram is congruent with — a, and its only infinity is congruent with 0, each being simple. Hence the m+l irreducible infinities of H (z) are congruent with a, GI, GZ, . . . , cm, and its /* + 1 irreducible zeros are congruent with 0, 71, 7s> •••>%*; and therefore m + 1 = p, + 1, 139.] OF SECONDARY FUNCTIONS 281 From the first it follows* that the number of infinities of a doubly-periodic function of the second kind in a parallelogram of periods is equal to the number of its zeros, and that the excess of the sum of the former over the sum of the latter is congruent with , (°>' i w i , + — , log it -. log u, - \7Tl TTl 6 ' / the sign being the same as that of 9t ( — \10) The result just obtained renders it possible to derive another expression for F (z), substantially due to Hermite. Consider a function F (Z) = °-Q-7i) 0-0-72).. -0-0-7™) ePZ (T(z-c1)(r(z-c2)...ar(z-cm) ' where p is a constant. Evidently F1 (z) has the same zeros and the same infinities, each in the same degree, as F (z). Moreover F, (Z + 2ft)) = Fl (Z) e2,(2C-2y) + 2pWj F1 (Z + 2ft)') = F! (Z) e2V(2e-2y)+2P«,'t If, then, we choose points c and 7, such that Sc — £7 = a, and we take p = \ where a and X are the constants of G (z), then F, (z + 2co) = ^ (z), F, (z + 2ft>') = n'Fj. (z). The function Fl (z)/F(z) is a doubly-periodic function of the first kind and by the construction of Fl (z) it has no zeros and no infinities in the finite part of the plane: it is therefore a constant. Hence F(z] = A gfr-'ftM*- •/»)•••* (*—*») ^ a(z- c,) a- (z - C2). . .o- (z - Cm) where Sc — £7 = a, and a and A, are determined as for the function G (z}. 140. One of the most important applications of secondary doubly-periodic functions is that which leads to the solution of Lame's equation in the cases when it can be integrated by means of uniform functions. This equation is subsidiary to the solution of the general equation, characteristic of the potential of an attracting mass at a point in free space; and it can be expressed either in the form jY = (Ak'2 sn2 z + B) w, or in the form - 2 - = (A@ (z) + B} w, * Frobenius, Crelle, xciii, pp. 55 — 68, a memoir which contains developments of the properties of the function G (z). The result appears to have been noticed first by Brioschi, (Comptes Ilendus, t. xcii, p. 325), in discussing a more limited form. 282 LAMP'S [140. according to the class of elliptic functions used. In order that the integral may be uniform, the constant A must be n (n -f 1), where n is a positive integer ; this value of A, moreover, is the value that occurs most naturally in the derivation of the equation. The constant B can be taken arbitrarily. The foregoing equation is one of a class, the properties of which have been established* by Picard, Floquet, and others. Without entering into their discussion, the following will suffice to connect them with the secondary periodic function. Let two independent special solutions be g (z) and h (z), uniform functions of z ; every solution is of the form ag (z} + /3h (z}, where a and /3 are constants. The equation is unaltered when z + 2w is substituted for z ; hence g {z + 2&>) and h (z + 2&>) are solutions, so that we must have g (z + 2w) = Ag (z} + Bh (z}, h(z + 2o>) = Cg (z) + Dh (z\ where, as the functions are determinate, A, B, C, D are determinate constants, such that AD — BC is different from zero. Similarly, we obtain equations of the form g (z + 2co') = A'g (z) + B'h (z\ h(z + 2co') = C'g (z} + D'h (z}. Using both equations to obtain g (z + 2o> + 2&/) in the same form, we have BC' = B'C, AB' + BD' = A'B + B'D ; and similarly, for h (z + 2w + 20)'), we have C G' A-D A'-U therefore -~ = -™ = o, — ~ — = — ™ — = e. x> -D n n Let a solution F (z} = ag (z) + bh (z) be chosen, so as to give if possible. The conditions for the first are a b so that a/b (= £) must satisfy the equation and the conditions for the second are aA' + bCf aB' + bD' * Picard, Comptes Rendus, t. xc, (1880), pp. 128—131, 293—295; Crelle, t. xc, (1880), pp. 281—302. Floquet, Comptes Rendus, t. xcviii, (1884), pp. 82 — 85 ; Ann. de VEc. Norm. Sup., 3mc Ser., t. i, (1884), pp. 181—238. 140.] DIFFERENTIAL EQUATION 283 so that £ must satisfy the equation A'-D'=^B'~~. These two equations are the same, being p.-«g-ft*a Let £j and £2 be the roots of this equation which, in general, are unequal ; and let fa, fa and fa, fa.' be the corresponding values of /z, //. Then two functions, say FI (z) and F^ (z), are determined : they are independent of one another, so therefore are g (z) and h (z) ; and therefore every solution can be expressed in terms of them. Hence a linear differential equation of the second order, having coefficients that are doubly-periodic functions of the first kind, can generally be integrated by means of doubly -periodic functions of the second kind. It therefore follows that Lame's equation, which will be taken in the form can be integrated by means of secondary doubly-periodic functions. 141. Let z = c be an accidental singularity of w of order m ; then, for points z in the immediate vicinity of c, we have and therefore 2mp ~ z- c + P°SltlVe P°wers °f * - Since this is equal to n (n + 1) @ (z) + B it follows that c must be congruent to zero and that m, a positive integer, must be n. Moreover, p = 0. Hence the accidental singularities of w are congruent to zero, and each is of order n. The secondary periodic function, which has no accidental singularities except those of order n congruent to z = 0, has n irreducible zeros. Let them be — alt — a2,..., — an; then the form of the function is Hence 1 *? = ,-»?« + or, taking p = - ^(ar), we have and therefore i *? - 1 (*?)' . n(> (,) - X f> (« + «, 19 O^ W2\dzj * v y r»i 284 INTEGRATION But, by Ex. 3, § 131, we have [141. 4 r=1 £> (ar) - p (z) , by Ex. 4, § 131. Thus W Now r=l S=l . g> (a.) - g> (ar) - g) («) ' g> (a,) - g> (^) 4^?3 (^r) - ^2ip Q) - #, + %>' (a*) & (a,) where g> (ar) - £> (a.) Let the constants a be such that (O - £> (a2) + -H...-0 /i equations of which only n — 1 are independent, because the sum of the n left-hand sides vanishes. Then iu the double summation the coefficient of i f .1 f u #>' (ar) — &' (z) . each of the tractions * )—,- — V\ is zero ; and so and therefore • -^-, = w (w + 1) p (z) + (2n — 1) 2 ^> (a,.). /IU GLZ" T=l Hence it follows that _<T(z + aJ <T(z + a2)...<r(z + an) -z?J("r) an (z} satisfies Lame's equation, provided the n constants a be determined by the preceding equations and by the relation B = (2n-l) I pfa). 141.] OF LAMP'S EQUATION 285 Evidently the equation is unaltered when — z is substituted for z ; and therefore is another solution. Every solution is of the form MF(z} + NF(-z), where M and N are arbitrary constants. COEOLLARY. The simplest cases are when n = l and n = 2. When n = 1, the equation is • j-r- + B : w dzz there is only a single constant a determined by the single equation B = p (a), and the general solution is ,, a (2 + a) ... ,ra(z — a] ... , w = M — ^- 7-^-/ e~2£(a) + N - ---- ' ' es^a> o- (z} a (z) When n = 2, the equation is -J-. = 6(0 (z} + B. w dz* The general solution is ^ where a and b are determined by the conditions Rejecting the solution a+b = 0, we have a and b determined by the equations p (a) For a full discussion of Lame's equation and for references to the original sources of information, see Halphen, Traite des fonctions elliptiques, t. ii, chap, xn., in particular, pp. 495 et seq. Ex. When Lamp's equation has the form 1 d?w - -T-5 =n (n + 1) £2 sn20 - h. w dz2 ^ obtain the solution for w = l, in terms of the Jacobian Theta-Functions, where co is determined by the equation dn2o> = A-F ; and discuss in particular the solution when h has the values l+£2, 1, £2. Obtain the solution for » = 2 in the form i +B - fe^)e- K& .1 J SI e(») j' 286 PSEUDO-PERIODIC FUNCTIONS [141. where X and w are given by the equations (2P sn2 a - 1 - F) (2F sn2 a - 1) (2 sn2 a - 1) 3Fsn4a-2(l+£2)sn2a + l ~ ' and a is derived from h by the relation Deduce the three solutions that occur when X is zero, and the two solutions that occur when X is infinite. (Hermite.) Doubly-Periodic Functions of the Third Kind. 142. The equations characteristic of a doubly-periodic function <I> (z) of the third kind are = <£(», <&(z + 2a)') = e~ »~Z Q(z), where m is an integer different from zero. Obviously the number of zeros in a parallelogram is a constant, as well as the number of infinities. Let a parallelogram, chosen so that its sides contain no zero and no infinity of <& (z}, have p, p + 2<w, p + 2&>' for three of its angular points; and let a1} a2, . .., a{ be the zeros and cl5 ..., cm be the infinities, multiplicity of order being represented by repetitions. Then using "^ (z) to denote , (log <£ (z)}, we have, as the equations characteristic of * and for points in the parallelogram where -ff (^) has no infinity within the parallelogram. Hence the integral being taken round the parallelogram : by using the Corollary to Prop. II. in § 116, we have 27ri (I - n) - - \ - \^L\ dz = Jp \ &> / so that I = n + m : or the algebraical excess of the number of irreducible zeros over the number of irreducible infinities is equal to in. z Again, since — = 1 + z — /A z — p, a c we have 2 2 h I — n = z"^ (z) — zH (z), z — a z — c and therefore 2-Tn (Sa — 2c) = jz*\? (z) dz, 142.] OF THE THIRD KIND 287 the integral being taken round the parallelogram. As before, this gives rp+2<a' rp+2<a < vnTri "I 2™ (2a - 2c) = 2ft)^ (z) dz - MV (z) - - (z + 2ft/) dz. Jp Jp ( ft) ) The former integral is rp+*»'(g) ,v x dz (*) miri for the side of the parallelogram contains* no zero and no infinity The latter integral, with its own sign, is <P(Z) ft) = 0 + {O + 2« + 2ft>')2 - (p + 2ft/)2} = 2TO7T* (p + ft) + 2ft)'). Hence 2a — Sc = m (&) + 2&/), giving the excess of the sum of the zeros over the sum of the infinities in any parallelogram chosen so as to contain the variable z and to have no one of its sides passing through a zero or an infinity of the function. These will be taken as the irreducible zeros and the irreducible infinities : all others are congruent with them. All these results are obtained through the theorem II. of § 116, which assumes that the argument of <y' is greater than the argument of &) or, what is the equivalent assumption (§ 129), that rjco' — w'co = ^iri. 143. Taking the function, naturally suggested for the present class by the corresponding function for the former class, we introduce a function a(z- d) <r(z- C2). ..<r(z — Cn) ' where the a's and the c's are connected by the relations Sa — Sc = m (&) + 2&>'), l—n = m. Then (f>(z) satisfies the equations characteristic of doubly-periodic functions of the third kind, if 0 = 4Xo) + 2ra77, k . 27rt = 4X&)2 + 2m?/ft) + 2/ift) + miri — Zmrj (&> + 2ft)') ; miri — 2mrj' (&> + 2ft)'), * Both in this integral and in the next, which contain parts of the form I — there is, as in J w Prop. VII., § 116, properly an additive term of the form 2iciri, where K is an integer ; but, as there, both terms can be removed by modification of the position of the parallelogram, and this modifi cation is supposed, in the proof, to have been made. 288 TERTIARY FUNCTIONS [143. k and k' being disposable integers. These are uniquely satisfied by taking with A; = 0, k' = m. Assuming the last two, the values of X and /JL are thus obtained so as to make <fr (z) a doubly-periodic function of the third kind. Now let Oj, ..., di be chosen as the irreducible zeros of <l> (z) and Ci, ..., cn as the irreducible infinities of <E> (2), which is possible owing to the conditions to which they were subjected. Then <3> (z)/<j> (z) is a doubly-periodic function of the first kind; it has no zeros and no infinities in the parallelogram of- periods and therefore none in the whole plane ; it is therefore a constant, so that 3> (z) = Ae"** "IZ*+^ - + (l|+8'')} ** <r(*-gi)°-(*-q»)-. •*(*-<*!) tr(z- d) <r(z- c.2)...o- (z - cn) ' a representation of <3> (z) in terms of known quantities. Ex. Had the representation been effected by means of the Jacobian Theta-Functions which would replace a (z) by H(z), then the term in z1 in the exponential would be absent. 144. No limitation on the integral value of m, except that it must not vanish, has been made : and the form just obtained holds for all values. Equivalent expressions in the form of sums of functions can be constructed : but there is then a difference between the cases of m positive and m negative. If m be positive, being the excess of the number of irreducible zeros over the number of irreducible infinities, the function is said to be of positive class m ; it is evident that there are suitable functions without any irreducible infinities — they are integral functions. When m is negative (= — n), the function is said to be of negative class n ; but there are no corresponding integral functions. 145. First, let m be positive. i. If the function have no accidental singularities, it can be expressed in the form A e**+i* a-(z — a1)a-(z — aa)...<r(z — am), with appropriate values of X and //.. ii. If the function have n irreducible accidental singularities, then it has m + n irreducible zeros. We proceed to shew that the function can be expressed by means of similar functions of positive class m, with a single accidental singularity. 145.] OF POSITIVE CLASS 289 Using X and /j, to denote , mri - 1 — ' and | - - + m (77 + 277'), &) a) which are the constants in the exponential factor common to all functions of the same class, consider a function, of positive class m with a single accidental singularity, in the form *m (z, u) = eW '' <r(u- 6X) o- (u - &„). • • <r (u - bm+1) <r(z-u)' where b1} b.2, ..., bm are arbitrary constants, of sum s, and m (&> + 2ft)') = 6OT+1 + fcj + b.> + . . . bm - u = bm+l +s-u. The function y-m satisfies the equations _mirzi y-w (z + 2<w, u) = i/rm (z, u), y,tt (z + 2&)', w) = e~' « -^m (z, u) ; regarded as a function of z, it has u for its sole accidental singularity, evidently simple. The function - — can be expressed in the form I\I/* I £ It I u — k) . . . a- (u — bm) o- {s — m (&) (r^-b,) ............ a-(z-bm) a{u- z-s + m(a> + 2~w7)} ' Regarded as a function of u, it has z, \, . . ., bm for zeros and z + s - m (to + 2o>') for its sole accidental singularity, evidently simple : also z + &J + ...+ bm - {z + s - m (&) + 2o/)} = m (w + 2o>'). Hence owing to the values of X arid p, it follows that -- } - x when re- f>m(*, tt) garded as a function of u, satisfies all the conditions that establish a doubly- periodic function of the third kind of positive class m, so that 1 1 ~i 7 ~ =r ^ and therefore mnz tym (z, u + 2o>) = ^m (z, u), ^m (z, u + 20)') = e~ijrm (z, u). Evidently -f m (z, u) regarded as a function of u is of negative class m : its infinities and its sole zero can at once be seen from the form -bm) o-{u-z-s+m(ca <r(u -z)*^-^)...*^- bm) a- {s - m (to + 2o)')j ' Each of the infinities is simple. In the vicinity of u = z, the expansion of the function is ^^z + positive integral powers of u — z : 19 290 TERTIARY FUNCTIONS [145. and, in the vicinity of u = br, it is C* ( 7\ r j + positive integral powers of u — br, Lv "~~ \Jrp where Gr (z) denotes r) <r(z-bi)--.<r(z-br-i)<r(z-br+l)...a(z-bm)o-{z+s-br-m(a>+2a>')} a- (br - 6j). ..cr (br - 6r_!) cr(br - br+l)...cr(br - bin) o-[s-ra(eo + 2&>')}' and is therefore an integral function of z of positive class m. Let 4> (14) be a doubly-periodic function of the third kind, of positive class m ; and let its irreducible accidental singularities, that is, those which occur in a parallelogram containing the point u, be a^ of order !+/*!, a., of order 1 + ju,2, and so on. In the immediate vicinity of a point ar, let --... ± \ - rr— r;r-... r-,-~- - -rr. cm du- du^J u — a,. Then proceeding as in the case of the secondary doubly-periodic functions (§ 137), we construct a function F(u) = 3?(u)^m(z, u). We at once have F (u + 2o>) = F (u) = F(u + 2a>'), so that F(u) is a doubly-periodic function of the first kind; hence the sum of its residues for all the poles in a parallelogram of periods is zero. For the infinities of F (u), which arise through the factor tym(z, u}, wea have as the residue for u = z -<*>(*), and as the residue for u = br, where r = 1, 2, ..., m, In the vicinity of a,., we have fyn (Z, u) = ^rm (Z, «r) + (u - Or) tym' (z, O.r) where dashes imply differentiation of ^rm {z, u} with regard to u, after which u is made equal to a,. ; so that in <I> (u) tym (z, u) the residue for u = ar, where r = l, 2, ..., is Er (z) = Ar ,jrm (z, ctr) + B, Tjrm' (z, a,.) + Cr tym" (z, ar) + ...+ Mr <^m^r) (z> ar\ Hence we have and therefore ®(z)= 2 E,(z)+ 2 <& (br) Gr(z), s=l r=l giving the expression of <l> (z) by means of doubly -periodic functions of tht third kind, which are of positive class m and have either no accidental singu-> larity or only one and that a simple singularity. 145.] OF NEGATIVE CLASS 291 The m quantities blt ..., bm are arbitrary; the simplest case which occurs is when the m zeros of &(z) are different and are chosen as the values of &!,..., bm. The value of 3>(z) is then <&(*)= 2 JS'.C*), s=l where the summation extends to all the irreducible accidental singularities ; while, if there be the further simplification that all the accidental singularities are simple, then <I> (z) = A1 TJrm (2, «!> + As tym (z, ot2) + . . ., the summation extending to all the irreducible simple singularities. The quantity tym (z, ar), which is equal to ) <r(z-bd...<r(z- bm) <r{z + 2b-m(<o + 2ft/) - ar] a-(ar — b1)...a- (ar - bm) <r {26 - m (co + 2ft>')} a- (z - ar) ' and is subsidiary to the construction of the function E (z\ is called the simple element of positive class m. In the general case, the portion gives an integral function of z, and the portion 2 Es (z) gives a fractional s=l function of z. 146. Secondly, let m be negative and equal to — n. The equations satisfied by & (z} are i = <I> 0), <I> (z + 2ft)') = e w <£ 0), and the number of irreducible singularities is greater by n than the number of irreducible zeros. One expression for <i> (z} is at once obtained by forming its reciprocal, which satisfies the equations 11 1 -2-** i f\ /K / -\ > and is therefore of the class just considered: the value of is of the q>(^) form ZEs(z) + ^ArGr(z}. For purposes of expansion, however, this is not a convenient form as it gives only the reciprocal of <I> (z}. To represent the function, Appell constructed the element TT sv°° Ffr-K»-*Wl 7r(2 gr— * . • cot — *- 19—2 292 TERTIARY FUNCTIONS [146. which, since the real part of to' fan is positive, converges for all values of z and y, except those for which z = y (mod. 2&>, 2&>'). For each of these values one term of the series, and therefore the series itself, becomes infinite of the first order. Evidently %„ (z, y + 2o>) = %M (z, y}, niryi Xn (z, y + 2eo') = e ° %„(*, y); therefore in the present case 0(y)=*(3f)jfr (**?)> regarded as a function of ^/, is a doubly-periodic function of the first kind. Hence the sum of the residues of its irreducible accidental singularities is zero. When the parallelogram is chosen, which includes z, these singularities are (i) y = z, arising through %n (z, y} ; (ii) the singularities of <£ (y}, which are at least n in number, and are n + I when <& has I irreducible zeros. The expansion of Xn 0> y), in powers of y - z, in the vicinity of the point z, is + positive integral powers of y — z ; y-z therefore the residue of II (y) is Let ctr be any irreducible singularity, and in the vicinity of a,, let 3> (y) denote d -I- positive integral powers of y — Or, where the series of negative powers is finite because the singularity is accidental ; then the residue of H (y} is Ar ^ (Z, Or) + Br Xn (*, «r) + Cr %,/' (z, Ct,) + . . . + Pr X*™ 0> «>')> where %n(A) (^, ar) is the value of dx%n (z, y) dy* when y = 0r after differentiation. Similarly for the residues of other singu larities : and so, as their sum is zero, we have <£ (Z) = 2 {Ar Xn (*, «r) + Br Xn (*, «•,) + ...+ P, XnW (?, «r)}, the summation extending over all the singularities. 146.] OF NEGATIVE CLASS 293 The simplest case occurs when all the N(>n) singularities a are accidental and of the first order ; the function 4> (z) can then be expressed in the form Al Xn (Z, «i) + A2 Xn (Z, Oj) + . . . + AN Xn (z, «#)• The quantity Xn (z, a), which is equal to T *^" ^p{(«-i)»'+«} TT 0 - a a 2/6 COt — -^. . 2(0 is called the simple element for the expression of a doubly-periodic function of the third kind of negative class n. Ex. Deduce the result _ ^ ( — iVcot TT snu s=-oov I 2K /' 147. The function Xn (z, y} can be used also as follows. Since Xm (z, y), qua function of y, satisfies the equations %m (z, 11 + 2(i)} = Y™ (z, 7/\ llv \ s {/ ' / /V//fc \ J ts /' miryi Xm (z, y + 2o/) = e~^xm (z, y), which are the same equations as are satisfied by a function of y of positive class m, therefore Xm (<*> z), which is equal to 2 e cot being a function of z, satisfies the characteristic equations of § 142 ; and, in the vicinity of z = a, Xm (a> z) — - — + positive integral powers of z — a. Z ~~" OC If then we take the function 4> (z) of § 145, in the case when it has simple singularities at alt «2, ... and is of positive class m, then 4> (z) + A, xw (a, , is a function of positive class m without any singularities: it is therefore equal to an integral function of positive class m, say to G(z)t where G (z) = Ae^+^a- (z-al}...(r(z- am), so that 3>(z) = G(z)-A1Xm(ct1,2)-A,xm(<Xt,z)-.... Ex. As a single example, consider a function of negative class 2, and let it have no zero within the parallelogram of reference. Then for the function, in the canonical product-form of § 143, the two irreducible infinities are subject to the relation and the function is * (z) = AV° "V" - o- (z— Cj) o- (z-c2)' 294 TERTIARY FUNCTIONS [147. The simple elements to express 3> (z) as a sum are 2.<!iri , , » {{s-lX + Cl} ,77, ' « *« " «rt (s-C! -2*,), 4iri, ,, 7T -(ci-<o)» - r-w-c'i TT = _e<-> 2 e a> cot — (2 + 0 j-2no) after an easy reduction, 4irj The residue of *(s) for cn which is a simple singularity, is 'Us-( Al = Ktfa v< and for c2, also a simple singularity, it is , so that ^- = -ew =-ew ^2 Hence the expression for 4> (z) as a sum, which is ! becomes Al (X2 (2, Cj) - e u ^2 (^ - ci)} that is, it is a constant multiple of Again, — j - - <r(z- GJ) a- (z + c^ - 2o>- on changing the constant factor. Hence it is possible to determine L so that •ni Tti " C' « c - e<a Taking the residues of the two sides for z=c1} we have and therefore finally we have -C]*- — Ci -- C, Le™ <» = e °> >-.•>-* TtlC (a (s, c) - e w X2 (2> - c) <* Cot^L(2-c1-2su)')-e w cot - - (z + cx - 2sw') K 2<a 2a> ) the right-hand side of which admits of further modification if desired. 147.] PSEUDO-PERIODIC FUNCTIONS 295 Many examples of such developments in trigonometrical series are given by Hermite*, Biehlerf, HalphenJ, Appell§, and Krause||. 148. We shall not further develop the theory of these uniform doubly- periodic functions of the third kind. It will be found in the memoirs of Appell§ to whom it is largely due; and in the treatises of Halphen**, and of Rausenberger"f"f. It need hardly be remarked that the classes of uniform functions of a single variable which have been discussed form only a small proportion of functions reproducing themselves save as to a factor when the variable is subjected to homographic substitutions, of which a very special example is furnished by linear additive periodicity. Thus there are the various classes of pseudo-automorphic functions, (§ 305) called Thetafuchsian by Pom- care, their characteristic equation being for all the substitutions of the group determining the function : and other classes are investigated in the treatises which have just been quoted. The following examples relate to particular classes of pseudo-periodic functions. Ex. 1. Shew that, if F (z) be a uniform function satisfying the equations m where b is a primitive mth root of unity, then F(z) can be expressed in the form where f(z) denotes the function and prove that \F(z)dz can be expressed in the form of a doubly-periodic function together with a sum of logarithms of doubly-periodic functions with constant coefficients. (Goursat.) * Comptes Rendus, t. Iv, (1862), pp. 11—18. t Sur les developpements en series des fonctions doublement periodiqucs de troisieme espece, (These, Paris, Gauthier-Villars, 1879). £ Traite des fonctions elliptiques, t. i, chap. xm. § Annales de VEc. Norm. Sup., 3rae S6r., t. i, pp. 135—164, t. ii, pp. 9—36, t. iii, pp. 9—42. || Math. Ann., t. xxx, (1887), pp. 425—436, 516—534. '* Traite des fonctions elliptiques, t. i, chap. xiv. ft Lehrbuch der Theorie der periodischen Functional, (Leipzig, Teubner, 1884), where further references are given. 296 PSEUDO-PERIODIC FUNCTIONS [148. Ex. 2. Shew that, if a pseudo-periodic function be denned by the equations and if, in the parallelogram of periods containing the point z, it have infinities c, ... such that in their immediate vicinity then/ (2) can be expressed in the form -'^'^«{^I+ ...... +«»,£}«—>, the summation extending over all the infinities of/ (z) in the above parallelogram of periods, and the constants (715 ... being subject to the condition + iVS Cl = A o>' — X'«o. Deduce an expression for a doubly-periodic function <f) (z) of the third kind, by assuming /W-f]8. (Halphen.) (f> \g) Ex. 3. If S(z) be a given doubly-periodic function of the first kind, then a pseudo-periodic function F(z), which satisfies the equations F(z + ^} = F(z), mriz F (z + 2o>') = e ~"~ S (z} F (z), where n is an integer, can be expressed in the form where -4 is a constant and TT (2) denotes the summation extending over all points &,. and the constants Br being subject to the relation Explain how the constants b, G and B can be determined. (Picard.) Ex. 4. Shew that the function F(z) defined by the equation for values of \z\, which are <1, satisfies the equation and that the function Fl(a!)=^ ^rjr-£i where (j)(,v) = 3? — 1, and </>„(.*•')> f()r positive and negative values of n, denotes (/> [0 {<£ 0 (#)}] <f> being repeated n times, and a is the positive root of a3 — a - 1 = 0 ; satisfies the equation for real values of the variable. Discuss the convergence of the series which defines the function Fl (x). (Appell.) CHAPTER XIII. FUNCTIONS POSSESSING AN ALGEBRAICAL ADDITION-THEOREM. 149. WE may consider at this stage an interesting set* of important theorems, due to Weierstrass, which are a justification, if any be necessary, for the attention ordinarily (and naturally) paid to functions belonging to the three simplest classes of algebraic, simply-periodic and doubly-periodic functions. A function <f> (u) is said to possess an algebraical addition theorem, when among the three values of the function for arguments u, v, and u + v, where u and v are general and not merely special arguments, an algebraical equation exists f having its coefficients independent of u and v. 150. It is easy to see, from one or two examples, that the function does not need to be a uniform function of the argument. The possibility of multiformity is established in the following proposition : A function defined by an algebraical equation, the coefficients of which are uniform algebraical functions of the argument, or are uniform simply -periodic functions of the argument, or are uniform doubly -periodic functions of the argument, possesses an algebraical addition-theorem. * They are placed in the forefront of Schwarz's account of Weierstrass's theory of elliptic functions, as contained in the Formeln und Lehrsdtze zum Gebrauche der elliptischen Functionen; but they are there stated (§§ 1—3) without proof. The only proof that has appeared is in a memoir by Phragmen, Acta Math., t. vii, (1885), pp. 33—42; and there are some statements (pp. 390—393) in Biermann's Theorie der analytischen Functionen relative to the theorems. The proof adopted in the text does not coincide with that given by Phragme'n. t There are functions which possess a kind of algebraical addition -theorem ; thus, for instance, the Jacobian Theta-functions are such that eA(u + w) O^ (u- v) can be rationally ex pressed in terms of the Theta-functions having it and v for their arguments. Such functions are, however, naturally excluded from the class of functions indicated in the definition. Such functions, however, possess what may be called a multiplication-theorem for multipli cation of the argument by an integer, that is, the set of functions 6 (nut) can be expressed algebraically in terms of the set of functions 6 (M). This is an extremely special case of a set of transcendental functions having a multiplication-theorem, which are investigated by Poincare, Liouville, 4°" S6r., t. iv, (1890), pp. 313—365. 298 EXAMPLES OF FUNCTIONS [150. First, let the coefficients be algebraical functions of the argument u. If the function defined by the equation be U, we have Umg0 (u) + Um~lgi (u) + ...+gm (u) = 0, where g0(u),gi(u}, ...,gm(u) are rational integral algebraical functions of u of degree, say, not higher than n. The equation can be transformed into un f/U\+ u'1-1/! ( U) + ... + fn ( U) = 0, where f0(U), fi(U), ••••> fn(U) are rational integral algebraical functions of U of degree not higher than m. If V denote the function when the argument is v, and W denote it when the argument is u + v, then w»/0 (7) + ^1/1 (7) + ... +fn (V) M 0, and (u + v)n/0 ( W) + (u + vY^f, ( W ) + . . . +fn ( W ) = 0. The algebraical elimination of the two quantities u and v between these three equations leads to an algebraical equation between the quantities /(£/"), /(7) and f (W), that is, to an algebraical equation between U, V, W, say of the form G(U, V, F) = 0, where G denotes an algebraical function, with coefficients independent of u and v. It is easy to prove that G is symmetrical in U and 7, and that its degree in each of the three quantities U, 7, W is wn2. The equation G = 0 implies that the function U possesses an algebraical addition- theorem. Secondly, let the coefficients* be uniform simply-periodic functions of the argument u. Let &> denote the period: then, by § 113, each of these TT'IL functions is a rational algebraical function of tan — . Let u' denote tan — ; then the equation is of the form Umg0 (u') + Um^g, (u'} + ...+ gm 00 = 0, where the coefficients g are rational algebraical (and can be taken as integral) functions of u'. If p be the highest degree of u' in any of them, then the equation can be transformed into u'vfo ( U) + u'P-1/! ( U) + . . . + fp ( U) = 0, where f0(U), fi(U), ..., fp(U) are rational integral algebraical functions of U of degree not higher than m. * The limitation to uniformity for the coefficients has been introduced merely to make the illustration simpler; if in any case they were multiform, the equation would be replaced by another which is equivalent to all possible forms of the first arising through the (finite) multiformity of the coefficients : and the new equation would conform to the specified conditions. 150.] POSSESSING AN ADDITION-THEOREM 299 Let v denote tan — , and w denote tan — -- ; then the corresponding cy &) values of the function are determined by the equations and w'*>f0(W) + w'p-*/! (W) + ... +fp (W) = 0. The relation between u', v', w' is u'v'w' + u' + v' - w' = 0. The elimination of the three quantities u', v', w' among the four equations leads as before to an algebraical equation G(U, V, W) = 0, where G denotes an algebraical function (now of degree mp'2) with coefficients independent of u and v. The function U therefore possesses an algebraical addition-theorem. Thirdly, let the coefficients be uniform doubly-periodic functions of the argument u. Let &> and &/ be the two periods ; and let @ (u), the Weier- strassian elliptic function in those periods, be denoted by £. Then every coefficient can be expressed in the form ~L ' where L, M, N are rational integral algebraical functions of f of finite degree. Unless each of the quantities N is zero, the form of the equation when these values are substituted for the coefficients is A+Bp'(u) = 0, so that A* = &(±?-g£-9*)\ and this is of the form Umff* (£) + U'^g, (|) + . . . + gm (£) - 0, where the coefficients g are rational algebraical (and can be taken as integral) functions of £ If q be the highest degree of £ in any of them, the equation can be transformed into where the coefficients / are rational integral algebraical functions of U of degree not higher than 2m. Let TJ denote $ (v) and f denote p(u + v); then the corresponding values of the function are determined by the equations ......... +fq(V)=0, By using Ex. 4, § 131, it is easy to shew that the relation between £, rj, £ is 300 WEIERSTRASS'S THEOREM ON FUNCTIONS [150. The elimination of £, ij, £ from the three equations leads as before to an algebraical equation G(U,V, W) = 0, of finite degree and with coefficients independent of u and v. Therefore in this case also the function U possesses an algebraical addition-theorem. If, however, all the quantities N be zero, the equation defining U is of the form Umh0 (£) + U^h, (£) + . . . + hm (£) = 0, and a similar argument then leads to the inference that U possesses an algebraical addition-theorem. The proposition is thus completely established. 151. The generalised converse of the preceding proposition now suggests itself : what are the classes of functions of one variable that possess an alge braical addition-theorem? The solution is contained in Weierstrass's theorem : — An analytical function <f> (u), which possesses an algebraical theorem, is either (i) an algebraical function of u ; or liru (ii) an algebraical function of e » , where w is a suitably chosen constant ; or (iii) an algebraical function of the elliptic function %>(u), the periods — or the invariants g.z and g3 — being suitably chosen constants. Let U denote </> (w). For a given general value of u, the function U may have m values where, for functions in general, there is not a necessary limit to the value of m ; it will be proved that, when the function possesses an algebraical addition- theorem, the integer m must be finite. For a given general value of U, that is, a value of U when its argument is not in the immediate vicinity of a branch-point if there be branch-points, the variable u may have p values, where p may be finite or may be infinite. Similarly for given general values of v and of V, which will be used to denote <£ (v). First, let p be finite. Then because u has p values for a given value of U and v has p values for a given value of V, and since neither set is affected by the value of the other function, the sum u + v has p2 values because any member of the set u can be combined with any member of the set v ; and this number p2 of values of u + v is derived for a given value of U and a given value of V. Now in forming the function <j>(u + v), which will be denoted by W, we have m values of W for each value of u + v and therefore we have mp2 values of W for the whole set, that is, for a given value of U and a given value of V. 151.] POSSESSING AN ADDITION-THEOREM 301 Hence the equation between U, V, W is of degree* mp2 in W, necessarily finite when the equation is algebraical ; and therefore m is finite. Because m is finite, U has a finite number m of values for a given value of u ; and, because p is finite, u has a finite number p of values for a given value of U. Hence U is determined in terms of u by an algebraical equation of degree m, the coefficients of which, are rational integral algebraical functions of degree p ; and therefore U is an algebraic function of u. 152. Next, let p be infinite ; then (see Note, p. 303) the system of values may be composed of (i) a single simply-infinite series of values or (ii) a finite number of simply-infinite series of values or (iii) a simply-infinite number of simply-infinite series of values, say, a single doubly-infinite series of values or (iv) a finite number of doubly-infinite series of values or (v) an infinite number of doubly-infinite series of values where, in (v), the infinite number is not restricted to be simply-infinite. Taking these alternatives in order, we first consider the case where the p values of u for a given general value of U constitute a single simply -infinite series. They may be denoted by f (u, n), where n has a simply-infinite series of values and the form of/ is such that f(u, 0) = u. Similarly, the p values of v for a given general value of V may be denoted by/(y, n), where n' has a simply-infinite series of values. Then the different values of the argument for the function W are the set of values given by f(u,n)+f(v,ri), for the simply-infinite series of values for n and the similar series of values for n'. The values thus obtained as arguments of W must all be contained in the series f(u + v, n"}, where n" has a simply-infinite series of values ; and, in the present case,/(w + w, n"} cannot contain other values. Hence for some values of n and some values of n', the total aggregate being not finite, the equation f(u,n}+f(v,n'}=f(u + v,n") must hold, for continuously varying values of u and v. In the first place, an interchange of u and v is equivalent to an interchange of n and n on the left-hand side; hence n" is symmetrical in n and n'. Again, we have df(u, n) _ df(u + v, n") du 3 (u + v) dv ' * The degree for special functions may be reduced, as in Cor. 1, Prop. XIII, § 118; but in no case is it increased. Similarly modifications, in the way of finite reductions, may occur in the succeeding cases ; but they will not be noticed, as they do not give rise to essential modification in the reasoning. 302 FORM OF ARGUMENT [152. so that the form of f(u, n) is such that its first derivative with regard to u is independent of u. Let 0 (n) be this value, where 0 (n), independent of u, may be dependent on n ; then, since we have f(u, n) = uO (n) + ty (n), -fy- (n) being independent of u. Substituting this expression in the former equation, we have the equation u6 (n) + ^ (n) + v9 (n'} + f (71') = (u + v)6 (n"} + ^ (n"), which must be true for all values of u and v ; hence e(n)=e(n") = d(n'), so that 6 (n) is a constant and equal to its value when n = 0. But when n is zero,/(w, 0) is u ; so that 9 (0) = 1 and ^ (0) = 0, and therefore f(u, n) = u + Tjr (n), where i/r vanishes with n. The equation defining ty is for values of n from a singly-infinite series and for values of n' from the same series, that series is reproduced for TO". Since ^ (n) vanishes with n, we take ^ (n) = HX (n), and therefore rc% (n) + n'% (n') = ri'x (n"). Again, when n' vanishes, the required series of values of n" is given by taking n" = n ; and, when n does not vanish, n" is symmetrical in n and n', so that we have n" = n + n' + nn\, where X is not infinite for zero or finite values of n or n'. Thus •HX (n) + n'x (n) = (n + TO' + -nw'X) % (w + ?*' + wi'X). Since the left-hand side is the sum of two functions of distinct and inde pendent magnitudes, the form of the equation shews that it can be satisfied only if X = 0, so that n" = n + n' ; and % 0) = % (n//) = %(n'\ so that each is a constant, say o> ; then f(u, n} = u + nco, which is the form that the series must adopt when the series f(u + v, n") is obtained by the addition of/(«, n) and/0, n')- 152.] IN A SIMPLY-INFINITE SERIES 303 It follows at once that the single series of arguments for W is obtained, as one simply-infinite series, of the form u + v+n"a). For each of these arguments we have m values of W, and the set of m values of W is the same for all the different arguments; that is, W has m values for a given value of U and a given value of V. Moreover, U has m values for each argument and likewise V; hence, as the equation between U, V, W is of a degree that is necessarily finite because the equation is algebraical, the integer m is finite. It thus appears that the function U has a finite number m of values for each value of the argument u, and that for a given value of the function the values of the argument form a simply-periodic series represented by u + nw. But the function tan ( — ) is such that, for a given value, the values of the V 03 J argument are represented by the series u + nw ; hence for each value of tan ( — 1 there are m values of U and for each value of U there is one value \ «o / of tan -- . It therefore follows, by SS 113, 114, that between U and tan (— } w \ to / there is an algebraical relation which is of the first degree in tan - - and the O) U rath degree in U, that is, U is an algebraic function of tan — - . Hence U is (I) an algebraic function also of e <" . Note. This result is based upon the supposition that the series of argu ments, for which a branch of the function has the same value, can be arranged in the form/(w, n), where n has a simply-infinite series of integral values. If, however, there were no possible law of this kind — the foregoing proof shews that, if there be one such law, there is only one such law, with a properly determined constant co — then the values would be represented by ul} u», ...,up with p infinite in the limit. In that case, there would be an infinite number of sets of values for u + v of the type WA + v^, where X and p might be the same or might be different ; each set would give a branch of the function W and then there would be an infinite number of values of W corresponding to one branch of U and one branch of V. The equation between U, V and W would be of infinite degree in W, that is, it would be transcendental and not algebraical. The case is excluded by the hypothesis that the addition-theorem is alge braical, and therefore the equation between U, V and W is algebraical. 153. Next, let there be a number of simply-infinite series of values of the argument of the function, say q, where q is greater than unity and may be either finite or infinite. Let ul} u.2, ..., uq denote typical members of each series. Then all the members of the series containing ul must be of the form 304 FORM OF ARGUMENT [153. fi (ui> n)> f°r an infinite series of values of the integer n. Otherwise, as in the preceding note, the sum of the values in the series of arguments u and of those in the same series of arguments v would lead to an infinite number of distinct series of values of the argument u + v, with a corresponding infinite number of values W ; and the relation between U, V, W would cease to be algebraical. In the same way, the members of the corresponding series containing ^ must be of the form/! (v1} ri) for an infinite series of values of the integer n'. Among the combinations the simply-infinite series fi(tii+v1} n") must occur for an infinite series of values of n"; and therefore, as in the preceding case, fi(uly n) = M1 + nw1, where toj is an appropriate constant. Further, there is only one series of values for the combination of these two series ; it is represented by Ui + v1 + n"wl. In the same way, the members of the series containing u2 can be repre sented in the form u2 + nco2, where o>2 is an appropriate constant, which may be (but is not necessarily) the same as Wj ; and the series containing u.2, when combined with the set containing v2, leads to only a single series represented in the form u.2 + v2 + ri'o)2. And so on, for all the series in order. But now since u2 + m2a)2, where m2 is an integer, is a value of u for a given value of U, it follows that U (u2 + ra2a>2) = U (w2) identically, each being equal to U. Hence U (M! + mlwl + 7n.,<y2) = U (i^ + ra^) = U (u^ = U, and therefore ^ + ml(al + ra2&>2 is also a value of u for the given value of U, leading to a series of arguments which must be included among the original series or be distributed through them. Similarly u1 + 2mr(i)r, where the coefficients ra are integers and the constants to are properly determined, represents a series of values of the variable u, included among the original series or distributed through them. And generally, when account is taken of all the distinct series thus obtained, the aggregate of values of the variable u can be represented in the form Wx+2wrtur, for \ — 1, 2, ..., K, where K is some finite or infinite integer. Three cases arise, (a) when the quantities « are equal to one another or can be expressed as integral multiples of only one quantity a>, (6) when the quantities &> are equivalent to two quantities f^ and O2 (the ratio of which is not real), so that each quantity &> can be expressed in the form a>r=plrfil+parsia> the coefficients plr, p2r being finite integers ; (c) when the quantities « are not equivalent to only two quantities, such as flj and fl2. 153.] SIMPLY-PERIODIC FUNCTIONS 305 For case (a), each of the K infinite series of values u can be expressed in the form u^+pci), for X = 1, 2, ..., « and integral values of p. First, let K be finite, so that the original integer q is finite. Then the values of the argument for W are of the type that is, MA + '?V +£>"&>, for all combinations of \ and fju and for integral values of p". There are thus K- series of values, each series containing a simply-infinite number of terms of this type. For each of the arguments in any one of these infinite series, W has ra values ; and the set of m values is the same for all the arguments in one and the same infinite series. Hence W has w/c2 values for all the arguments in all the series taken together, that is, for a given value of U and a given value of V. The relation between U, V, W is therefore of degree m«2, necessarily finite when the equation is algebraical ; hence m is finite. It thus appears that the function U has a finite number m of values for each value of the argument u, and that for a given value of the function there are a finite number K of distinct series of values of the argument of the form 7TU u+poi), w being the same for all the series. But the function tan -- has one value for each value of u and the series u+pat represents the series of 7TU values of u for a given value of tan — . It therefore follows that there are CO m values of U for each value of tan — and that there are K values of tan — to o> for each value of U ; and therefore there is an algebraical relation between U and tan — , which is of degree K in the latter and of degree m in the &) iiru TTlI former. Hence U is an algebraic function of tan — and therefore also of e M . Next, let K be infinite, so that the original integer q is infinite. Then, as in the Note in § 152, the equation between U, V, W will cease to be algebraical unless each aggregate of values u^+pw, for each particular value of p and for the infinite sequence X= 1, 2, ..., K, can be arranged in a system or a set of systems, say a in number, each of the form fp(u+pa), pp) for an infinite series of values of pp. Each of these implies a series of values fp(v+p'u>, pp) of the argument of V for the same series of values of pp as of pp> and also a series of values fp(u + v+p"(o, pp") of the argument of W for the same series of values of pp". By proceeding as in § 152, it follows that fp (u +pa>, pp} = u+pto +pp(0p, where &>p' is an appropriate constant, the ratio of which to &> can be proved F. 20 306 FORM OF ARGUMENT [153. (as in § 106) to be not purely real, and pp has a simply-infinite succession of values. The integer a may be finite or it may be infinite. When ay and all the constants o>' which thus arise are linearly equivalent to two quantities f^ and O2, so that the terms additive to u can be expressed in the form 8^ + s.2fl», then the aggregate of values u can be expressed in the form for a simply-infinite series for pl and for p2 ; and p has a series of values 1, 2, ..., <r. This case is, in effect, the same as case (6). When o) and all the constants «' are not linearly equivalent to only two quantities, such as Oj and IL>, we have a case which, in effect, is the same as case (c). These two cases must therefore now be considered. For case (6), either as originally obtained or as derived through parfc of case (a), each of the (doubly) infinite series of values of u can be expressed in the form for X = 1, 2, ..., <r and for integral values of _p, and p,. The integer a may be finite or infinite ; the original integer q is infinite. First, let cr be finite. Then the values of the argument for W are of the type that is, u\ + v^ +pi"£li + p2"O2, for all combinations of \ and p and for integral values of £>/' and p.". There are thus cr2 series of values, each series containing a doubly-infinite number ofl terms of this type. For every argument there are m values of W ; and the set of m values is the same for all the arguments in one and the same infinite series. Thus W has mo-2 values for all the arguments in all the series, that is, for a given value of U and a given value of V; and it follows, as before, from the consideration i of the algebraical relation, that m is finite. The function U thus has m values for each value of the argument u ; and for a given value of the function there are cr series of values of the argument, each series being of the form wx + PI^I +p.2Q*- Take a doubly-periodic function © having Oj and H2 for its periods, such*1 that for a given value of © the values of its arguments are of the foregoing form. Whatever be the expression of the function, it is of the order cr. , Then U has m values for each value of @, and @ has one value for each'. value of U; hence there is an algebraical equation between U and ©, ow * All that is necessary for this purpose is to construct, by the use of Prop. XII, § 118, ai function having, as its irreducible simple infinities, a series of points aj, a2,..., a<7 — special* values of «j, w2, ..., ua— in the parallelogram of periods, chosen so that no two of the <r points a coincide. 153.] DOUBLY-PERIODIC FUNCTIONS 307 :he first degree in the latter and of the rath degree in U: that is, U is an algebraical function of @. But, by Prop. XV. § 119, © can be expressed in the form where L, M, N are rational integral algebraical functions of $ (u), if f^ and H2 be the periods of g) (u); and g)' (u) is a two- valued algebraical function of jjp (u), so that © is an algebraical function of i@ (u). Hence also U is an algebraical function of $(u\ the periods o/<p (u) being properly chosen. This inference requires that a, the order of ©, be greater than 1. Because U has m values for an argument u, the symmetric function St/" has one value for an argument u and it is therefore a uniform function. But each term of the sum has the same value for u+pifli+pflt as for u ; and therefore this uniform function is doubly-periodic. The number of independent doubly-infinite series of values of u for a uniform doubly- periodic function is at least two : and therefore there must be at least two doubly-infinite series of values of u, so that <r > 1. Hence a function, that possesses an addition-theorem, cannot have only one doubly-infinite series of values for its argument. If cr be infinite, there is an infinite series of values of u of the form + p^ + p.flz ; an argument, similar to that in case (a), shews that this is, in effect, the same as case (c). It is obvious that cases (ii), (iii) and (iv) of § 152 are now completely covered ; case (v) of § 152 is covered by case (c) now to be discussed in § 154. 154. For case (c), we have the series of values u represented by a number of series of the form where the quantities &> are not linearly equivalent to two quantities flj and Q2- The original integer q is infinite. Then, by §§ 108, 110, it follows that integers m can be chosen in an unlimited variety of ways so that the modulus of r=l is infinitesimal, and therefore in the immediate vicinity of any point u^ there is an infinitude of points at which the function resumes its value. Such a function would, as in previous instances, degenerate into a mere constant ; and therefore the combination of values which gives rise to this case does not occur. All the possible cases have been considered: and the truth of Weierstrass's 20—2 308 EXAMPLES [154. theorem* that a function, which has an algebraical addition-theorem, is either imi an algebraical function of u, or of e " (where &> is suitably chosen), or of g> (u), where the periods of @(u) are suitably chosen, is established; and it has incidentally been established — it is, indeed, essential to the derivation of the theorem — that a function, which has an algebraical addition-theorem, has only a finite number of values for a given argument. It is easy to see that the first derivative has only a finite number of values for a given argument; for the elimination of U between the algebraical equations , , leads to an equation in U' of the same finite degree as G in U. Further, it is now easy to see that if the analytical function <£ (u), which possesses an algebraical addition-theorem, be uniform, then it is a rational iiru function either of u, or of e w , or of $> (u) and $' (u) ; and that any uniform function, which is transcendental in the sense of § 47 and which possesses an algebraical addition-theorem, is either a simply-periodic function or a doubly- periodic function. The following examples will illustrate some of the inferences in regard to the number of values of <p (u + v) arising from series of values for u and v. Ex. I. Let U=u* + (2u+l)*. Evidently m, the number of values of U for a value of u, is 4 ; and, as the rationalised form of the equation is the value of p, being the number of values of u for a given value of U, is 2. Thus the equation in W should be, by § 151, of degree (4.22 — ) 16. This equation is n {3 ( W2 - U2 - F2) + 1 - 2kr} = 0, HI where kr is any one of the eight values of W(2W*-I)*+U(2U*-l$+V(2V*-l)*; ' • an equation, when rationalised, of the 16th degree in W. Ex. 2. Let U=cosu. Evidently m = l; the values of u for a given value of U are contained in the double series u + 2irn, -u + 2irn, for all values of n from -QO to +GO. The values of u + v are , that is, u + v + 27rp; -u + 27rn+v + 2irm, that is, -u + v + 2-n-p ; , that is, u-v + ^Trp; -u + 2irn-v + 2irm, that is, -u-v + Znp, * The theorem has been used by Schwarz, Ges. Werke, t. ii, pp. 260—268, in determining all the families of plane isothermic cirrves which are algebraical curves, an 'isothermic' curve being of the form u = c, where w is a function satisfying the potential-equation 154.] THE DIFFERENTIAL EQUATION 309 to that the number of series of values of u+v is four, each series being simply-infinite. It might thus be expected that the equation between U, V, W would be of degree 4 = ) 4 in W ; but it happens that cos (u + v)=cos( -u-v), and so the degree of the equation in W is reduced to half its degree. The equation is W2 - 2 WU V+ U2 + V2 - 1 = 0. Ex. 3. Let U=&iiu. Evidently m = l; and there are two doubly-infinite series of values of u determined by a given value of U, having the form u + 2ma> + 2m'<o', o> - w + 2mo> + 2m V. Hence the values of u + v are = u+v (mod. 2c0, 2o>') ; = ca-u + v (mod. 2«, 2«') ; = ca + u-v(mod. 2o>, 2<o') ; = -u-v (mod. 2o>, 2&>') ; four in number. The equation may therefore be expected to be of the fourth degree in W; it is 4 (1 - 6T2) (1 - F2) (1 - IF2) = (2 - U2- F2- IF2 +£2*7272 W2^ 155. But it must not be supposed that any algebraical equation between U, V, W, which is symmetrical in U and V, is one necessarily implying the representation of an algebraical addition-theorem. Without entering into a detailed investigation of the formal characteristics of the equations that are suitable, a latent test is given by implication in the following theorem, also due to Weierstrass : — If an analytical function possess an algebraical addition-theorem, an algebraical equation involving the function and its first derivative with regard to its argument exists ; and the coefficients in this equation do not involve the argument of the function. The proposition might easily be derived by assuming the preceding proposition, and applying the known results relating to the algebraical dependence between those functions, the types of which are suited to the representation of the functions in question, and their derivatives ; we shall, however, proceed more directly from the equation expressing the algebraical addition-theorem in the form G(U,V, F) = 0, which may be regarded as a rationally irreducible equation. Differentiating with regard to u, we have WU'+MW^Q dUL +dW ' and similarly, with regard to v, we have a>+ *<=<>, from which it follows that 310 EXPRESSION OF [155. This equation* will, in general, involve W; in order to obtain an equation free from W, we eliminate W between n A a ^^ rr/ d6r Tr/ G = 0 and ^j- U' = „ V , oil ov the elimination being possible because both equations are of finite degree; and thus in any case we have an algebraical equation independent of W and involving U, U', V, V. Not more than one equation can arise by assigning various values to v, a quantity that is independent of u ; for we should have either inconsistent equations or simultaneous equations which, being consistent, determine a! limited number of values of U and U' for all values of u, that is, only a number of constants. Hence there can be only one equation, obtained by assigning varying values to v; and this single equation is the algebraical equation between the function and its first derivative, the coefficients being independent of the argument of the function. Note. A test of suitability of an algebraical equation G — 0 between three variables U, V, W to represent an addition-theorem is given by the condition that the elimination of W between G-Q and U'^-V — dU~ dV leads to only a single equation between U and U' for different values of V and V. Ex. Consider the equation (Z-U- V- W)*-4(1-U}(1- F)(l- F) = 0. The deduced equation involving U1 and V is (2FTF- V- W+ U} U' = (2UW- U- W+ V) V, , th-it W (V-U}(V'+U'} = (SV~lTUr The elimination of W is simple. We have _ (27-1) U'-(2U-\) F" F U'-l-U V' utd 2 U V W-« ( Neglecting 4 (F+ U— 1) = 0, which is an irrelevant equation, arid multiplying by (2F— 1) U' — (2U—l) F', which is not zero unless the numerator also vanish, and this would make both U' and V zero, we have ( F+ U- 1) {(1 - F) U' - (1 - U} F'} 2 = (1 - U) (1 - F) ( U' - F') (2 F- 1) U' - (2 U- 1) F'}, and therefore V(U-V}(1- V] (7'2+ U( F- U} (1 - U} F'2 = 0. It is permissible to adopt any subsidiary irrational or non-algebraical form as the equivalent of G = 0, provided no special limitation to the subsidiary form be implicitly adopted. Thus, if W can be expressed explicitly in terms of U and F, this resoluble (but irrational) equivalent of the equation often leads rapidly to the equation between U and its derivative. 155.] THE ADDITION-THEOREM 311 When the irrelevant factor U- V is neglected, this equation gives U'* F'2 U(l-U}~ V(l - V) ' the equation required : and this, indeed, is the necessary form in which the equation involving U and U' arises in general, the variables being combined in associate pairs. Each side is evidently a constant, say 4a2 ; and then we have Then the value of U is sin2 (aM+/3), the arbitrary additive constant of integration being /3 ; by substitution in the original equation, (3 is easily proved to be zero. 156. Again, if the elimination between a - o and — U' - — V aduu ~wv be supposed to be performed by the ordinary algebraical process for finding o/~y o/^r the greatest common measure of G and U' %Tf — V %-\r> regarded as functions of W, the final remainder is the eliminant which, equated to zero, is the differential equation involving U, U', V, F'; and the greatest common measure, equated to zero, gives the simplest equation in virtue of which the equations G = 0 and ^y U' = _-^ V subsist. It will be of the form oil ov f(W,U,V, U',V') = 0. If the function have only one value for each value of the argument, so that it is a uniform function, this last equation can give only one value for W', for all the other magnitudes that occur in the equation are uniform functions of their respective arguments. Since it is linear in W, the equation can be expressed in the form W = R(U, V, U', V'\ where R denotes a rational function. Hence* : — A uniform analytical function (f> (u), which possesses an algebraical addition-theorem, is such that (f> (u + v) can be expressed rationally in terms of $ (u), <£' (w), $ (v) and <j> (v). It need hardly be pointed out that this result is not inconsistent with the fact that the algebraical equation between (£ (u + v), (f> (u) and <f> (v) does not, in general, express $(u + v) as a rational function of (f> (u) and <f>(v). And it should be noticed that the rationality of the expression of <£ (u + v) in terms of <j) (u), $ (v), (/>' (w), $ (v) is characteristic of functions with an algebraical addition-theorem. Instances do occur of functions such that <j)(u + v) can be expressed, not rationally, in terms of <£ (u), </> (v), </>' (u), </>' (v) ; they do not possess an algebraical addition-theorem. Such an instance is furnished by %(u)', the expression of £(u + v), given in Ex. 3 of § 131, can be modified so ' as to have the form indicated. * The theorem is due to Weierstrass ; see Schwarz, § 2, (I.e. in note to p. 297). CHAPTER XIV. CONNECTION OF SURFACES. 157. IN proceeding to the discussion of multiform functions, it was stated (§ 100) that there are two methods of special importance, one of which is the development of Cauchy's general theory of functions of complex vari ables and the other of which is due to Riemann. The former has been explained in the immediately preceding chapters ; we now pass to the consideration of Riemann's method. But, before actually entering upon it, there are some preliminary propositions on the connection of surfaces which must be established ; as they do not find a place in treatises on geometry, an outline will be given here but only to that elementary extent which is necessary for our present purpose. In the integration of meromorphic functions, it proved to be convenient to exclude the poles from the range of variation of the variable by means of infinitesimal closed simple curves, each of which was thereby constituted a limit of the region : the full boundary of the region was composed of the aggregate of these non-intersecting curves. Similarly, in dealing with some special cases of multiform functions, it proved convenient to exclude the branch-points by means of infinitesimal curves or by loops. And, in the case of the fundamental lemma of § 16, the region over which integration extended was considered as one which possibly had several distinct curves as its complete boundary. These are special examples of a general class of regions, at all points within the area of which the functions considered are monogeiiic, finite, and continuous and, as the case may be, uniform or multiform. But, important as are the classes of functions which have been considered, it is necessary to consider wider classes of multiform functions and to obtain the regions which are appropriate for the representation of the variation of the variable in each case. The most conspicuous examples of such new functions are the algebraic functions, adverted to in §§ 94 — 99 ; and it is chiefly in view of their value and of the value of functions dependent upon them, as well as of the kind of surface on which their variable can be simply represented, that we now proceed to establish some of the topological properties of surfaces in general. 158. A surface is said to be connected when, from any point of it to any other point of it, a continuous line can be drawn without passing out of the 158.] EXAMPLES OF CONNECTED SURFACES 313 surface. Thus the surface of a circle, that of a plane ring such as arises in Lambert's Theorem, that of a sphere, that of an anchor-ring, are connected surfaces. Two non-intersecting spheres, not joined or bound together in any manner, are not a connected surface but are two different connected surfaces. It is often necessary to consider surfaces, which are constituted by an aggregate of several sheets ; but, in order that the surface may be regarded as connected, there must be junctions between the sheets. One of the simplest connected surfaces is such a plane area as is enclosed and completely bounded by the circumference of a circle. All lines drawn in it from one internal point to another can be deformed into one another ; any simple closed line lying entirely within it can be deformed so as to be evanescent, without in either case passing over the circumference ; and any simple line from one point of the circumference to another, when regarded as an impassable barrier, divides the surface into two portions. Such a surface is called* simply connected. The kind of connected surface next in point of simplicity is such a plane area as is enclosed between and is completely bounded by the circumferences of two concentric circles. All lines in the surface from one point to another cannot necessarily be deformed into one another, e.g., the lines z0az and zj)z; a simple closed line cannot necessarily be deformed so as to be evanescent without crossing the boundary, e.g., the line az^bza ; and a simple line from a point in one part of the boundary to a point in another and different part of the boundary, such as a line AB, does not divide the surface into two portions but, set as an impassable barrier, it makes the surface simply connected. Again, on the surface of an anchor-ring, a closed line can be drawn in two essentially distinct ways, abc, cib'c', such that neither can be deformed so as to be evanes cent or so as to pass continuously into the other. If abc be made the only impassable barrier, a line such as afty cannot be deformed so as to be evanescent ; if ab'c' be made the only impassable barrier, the same holds of a line such as a/3'y'. In order to make the surface simply connected, two impassable barriers, such as abc and ab'c', must be set. Surfaces, like the flat ring or the anchor- Fig. 35. Fig. 36. * Sometimes the term vionadelphic is used. The German equivalent is einfach ziisammen- hangend. 314 CROSS-CUTS AND LOOP-CUTS [158. ring, are called* multiply connected] the establishment of barriers has made it possible, in each case, to modify the surface into one which is simply connected. 159. It proves to be convenient to arrange surfaces in classes according to the character of their connection ; and these few illustrations suggest that the classification may be made to depend, either upon the resolution of the surface, by the establishment of barriers, into one that is simply connected, or upon the number of what may be called independent irreducible circuits. The former mode — that of dependence upon the establishment of barriers — will be adopted, thus following Biemann-f- ; but whichever of the two modes be adopted (and they are not necessarily the only modes) subsequent de mands require that the two be brought into relation with one another. The most effective way of securing the impassability of a barrier is to suppose the surface actually cut along the line of the barrier. Such a section of a surface is either a cross-cut or a loop-cut. If the section be made through the interior of the surface from one point Fig. 37. of the boundary to another point of the boundary, without intersecting itself or meeting the boundary save at its extremities, it is called a cross-cut\. Every part of it, as it is made, is to be regarded as boundary during the formation of the remainder ; and any cross-cut, once made, is to be regarded as boundary during the formation of any cross-cut subsequently made. Illustrations are given in Fig. 37. The definition and explanation imply that the surface has a boundary. Some surfaces, such as a complete sphere and a complete anchor-ring, do not possess a boundary; but, as will be seen later (§§ 163, 168) from the discussion of the evanescence of circuits, it is desirable to assign some boundary in order to avoid merely artificial difficulties as to the numerical * Sometimes the term polyadc.lphic is used. The German equivalent is mehrfach zusammen- Mngcnd. t " Grundlagen fur eine allgemeine Theorie der Functionen einer veriindeiiichen complexen Grosse," Eiemann's Gesammelte Werke, pp. 9 — 12; "Theorie der Abel'schen Functionen," ib.,/ pp. 84—89. When reference to either of these memoirs is made, it will be by a citation "et ih^ page or pages in the volume of lliemann's Collected Works. £ This is the equivalent used for the German word Querschnitt ; French writers use Section, and Italian writers use Trasversale or Taglio trasversale. 159.] CONNECTION DEFINED 315 expression of the connection. This assignment usually is made by taking for the boundary of a surface, which otherwise has no boundary, an infinitesimal closed curve, practically a point; thus in the figure of the anchor-ring (Fig. 36) the point a is taken as a boundary, and each of the two cross-cuts begins and ends in a. If the section be made through the interior of the surface from a point not on the boundary and, without meeting the boundary or crossing itself, return to the initial point, (so that it has the form of a simple curve lying Fig. 38. entirely in the surface), it is called* a loop-cut. Thus a piece can be cut out of a bounded spherical surface by a loop-cut (Fig. 38) ; but it does not necessarily give a separate piece when made in the surface of an anchor-ring. It is evident that both a cross-cut and a loop-cut furnish a double boundary-edge to the whole aggregate of surface, whether consisting of two pieces or of only one piece after the section. Moreover, these sections represent the impassable barriers of the pre liminary explanations ; and no specified form was assigned to those barriers. It is thus possible, within certain limits, to deform a cross-cut or a loop-cut continuously into a closely contiguous and equivalent position. If, for instance, two barriers initially coincide over any finite length, one or other can be slightly deformed so that finally they intersect only in a point ; the same modification can therefore be made in the sections. The definitions of simple connection and of multiple connection will nowf* be as follows : — A surface is simply connected, if it be resolved into two distinct pieces by every cross-cut; but if there be any cross-cut, which does not resolve it into distinct pieces, the surface is multiply connected. 160. Some fundamental propositions, relating to the connection of surfaces, may now be derived. * This is the equivalent used for the German word Riickkehrsclmitt ; French writers use the word Retroscction. t Other definitions will be required, if the classification of surfaces be made to depend on methods other than resolution by sections. 316 RESOLUTION BY CROSS-CUTS [160. I. Each of the two distinct pieces, into which a simply connected surface S is resolved by a cross-cut, is itself simply connected. If either of the pieces, made by a cross-cut ab, be not simply connected, then some cross-cut cd must be possible which will not resolve that piece into distinct portions. If neither c nor d lie on ab, then the obliteration of the cut ab will restore the original surface 8, which now is not resolved by the cut cd into distinct pieces. If one of the extremities of cd, say c, lie on ab, then the obliteration of the portion cb will change the two pieces into a single piece which is the original surface 8; and 8 now has a cross-cut acd, which does not resolve it into distinct pieces. If both the extremities lie on ab, then the obliteration of that part of ab which lies between c and d will change the two pieces into one ; this is the original surface 8, now with a cross-cut acdb, which does not resolve it into distinct pieces. These are all the possible cases should either of the distinct pieces of 8 not be simply connected ; each of them leads to a contradiction of the simple connection of 8', therefore the hypothesis on which each is based is untenable, that is, the distinct pieces of 8 in all the cases are simply connected. COROLLARY 1. A singly connected surface is resolved by n cross-cuts into Ti+1 distinct pieces, each simply connected; and an aggregate of m simply connected surfaces is resolved by n cross-cuts into n -f m distinct pieces each simply connected. COROLLARY 2. A surface that is resolved into two distinct simply con nected pieces by a cross-cut is simply connected before the resolution. COROLLARY 3. // a multiply connected surface be resolved into two different pieces by a cross-cut, both of these pieces cannot be simply connected. We now come to a theorem* of great importance : — II. If a resolution of a surface by m cross-cuts into n distinct simply connected pieces be possible, and also a different resolution of the same surface by fjb cross-cuts into v distinct simply connected pieces, then m — n = fj, — v. Let the aggregate of the n pieces be denoted by 8 and the aggregate of the v pieces by 2 : and consider the effect on the original surface of a united system of in + p simultaneous cross-cuts made up of the two systems of the m and of the /j, cross-cuts respectively. The operation of this system can be carried out in two ways : (i) by effecting the system of /u, cross-cuts on 8 and * The following proof of this proposition is substantially due to Neumann, p. 157. Another proof is given by Riemann, pp. 10, 11, and is amplified by Durege, Elemente der Theorie der Functional, pp. 183 — 190 ; and another by Lippich, see Durege, pp. 190 — 197. 160.] CONNECTIVITY 317 (ii) by effecting the system of m cross-cuts on 2 : with the same result on the original surface. After the explanation of § 159, we may justifiably assume that the lines of the two systems of cross-cuts meet only in points, if at all : let 8 be the number of points of intersection of these lines. Whenever the direction of a cross-cut meets a boundary line, the cross-cut terminates ; and if the direction continue beyond that boundary line, that produced part must be regarded as a new cross-cut. Hence the new system of /u, cross-cuts applied to S is effectively equiva lent to (j, + & new cross-cuts. Before these cuts were made, S was composed of n simply connected pieces ; hence, after they are applied, the new arrange ment of the original surface is made up of n + (/j, + 8) simply connected pieces. Similarly, the new system of m cross-cuts applied to 2 will give an arrangement of the original surface made up of v + (m + 8) simply connected pieces. These two arrangements are the same : and therefore n + fj, + 8 — v + in + 8, so that m — n = p — v. It thus appears that, if by any system of q cross-cuts a multiply connected surface be resolved into a number p of pieces distinct from one another and all simply connected, the integer q — p is independent of the particular system of the cross-cuts and of their configuration. The integer q—p is therefore essentially associated with the character of the multiple connection of the surface : and its invariance for a given surface enables us to arrange surfaces according to the value of the integer. No classification among the multiply connected surfaces has yet been made : they have merely been defined as surfaces in which cross-cuts can be made that do not resolve the surface into distinct pieces. It is natural to arrange them in classes according to the number of cross cuts which are necessary to resolve the surface into one of simple connection or a number of pieces each of simple connection. For a simply connected surface, no such cross-cut is necessary: then q = 0, p=l, and in general q — p = — l. We shall say that the connectivity* is unity. Examples are furnished by the area of a plane circle, and by a spherical surface with one hole^. A surface is called doubly- connected when, by one appropriate cross-cut, the surface is changed into a single surface of simple connection : then q = 1, p = 1 for this particular resolution, and therefore in general, q—p = Q. We * Sometimes order of connection, sometimes adelphic order ; the German word, that is used, is Grundzahl. + The hole is made to give the surface a boundary (§ 163). 318 EFFECT OF CROSS-CUTS [160. shall say that the connectivity is 2. Examples are furnished by a plane ring and by a spherical surface with two holes. A surface is called triply-connected when, by two appropriate cross-cuts, the surface is changed into a single surface of simple connection : then q = 2, p = l for this particular resolution and therefore, in general, q — p = l. We shall say that the connectivity is 3. Examples are furnished by the surface of an anchor- ring with one hole in it*, and by the surfaces -f- in Figure 39, the surface in (2) not being in one plane but one part beneath another. Fig. 39. And, in general, a surface will be said to be ^V-ply connected or its connectivity will be denoted by N, if, by N — 1 appropriate cross-cuts, it can be changed into a single surface that is simply connected |. For this particular resolution q = N—\, p = l: and therefore in general q-p = N-2, or N = q-p + 2. Let a cross-cut I be drawn in a surface of connectivity N. There are two cases to be considered, according as it does not or does divide the surface into distinct pieces. First, let the surface be only one piece after I is drawn : and let its connectivity then be N'. If in the original surface q cross-cuts (one of which can, after the preceding proposition, be taken to be I) be drawn dividing the surface into p simply connected pieces, then N = q-p+ 2. To obtain these p simply connected pieces from the surface after the cross-cut I, it is evidently sufficient to make the q — 1 original cross-cuts other than I ; that is, the modified surface is such that by q — 1 cross-cuts it is resolved into p simply connected pieces, and therefore Hence N' = N — 1, or the connectivity of the surface is diminished by unity. * The hole is made to give the surface a boundary (§ 163). t Riemann, p. 89. J A few writers estimate the connectivity of such a surface as N- 1, the same as the number of cross-cuts which can change it into a single surface of the simplest rank of connectivity : the estimate in the text seems preferable. 160.] ON THE CONNECTIVITY 319 Secondly, let the surface be two pieces after I is drawn, of connectivities Ni and N2 respectively. Let the appropriate JVj — 1 cross-cuts in the former, and the appropriate N2 — 1 in the latter, be drawn so as to make each a simply connected piece. Then, together, there are two simply connected pieces. To obtain these two pieces from the original surface, it will suffice to make in it the cross-cut I, the Ni — I cross-cuts, and the N2—l cross-cuts, that is, 1 + (Ni. — 1) + (N* — 1) or Nj, + N2 — 1 cross-cuts in all. Since these, when made in the surface of connectivity N, give two pieces, we have and therefore If one of the pieces be simply connected, the connectivity of the other is JV; so that, if a simply connected piece of surface be cut off a multiply connected surface, the connectivity of the remainder is unchanged. Hence : III. If a cross-cut be made in a surface of connectivity N and if it do not divide it into separate pieces, the connectivity of the modified surface is N—l; but if it divide the surface into two separate pieces of connectivities N! and N«, then Nl + N2 = N+ 1. Illustrations are shewn, in Fig. 40, of the effect of cross-cuts on the two surfaces in Fig. 39. IV. In the same way it may be proved that, if s cross-cuts be made in a surface of connectivity N and divide it into r+l separate pieces (where r^.s) of connectivities N1} N2, ..., Nr+l respectively, then a more general result including both of the foregoing cases. Thus far we have been considering only cross-cuts : it is now necessary to consider loop-cuts, so far as they affect the connectivity of a surface in which they are made. 320 EFFECT OF LOOP-CUTS [160. A loop-cut is changed into a cross-cut, if from A any point of it a cross-cut be made to any point C in a boundary-curve of the original surface, for CAbdA (Fig. 41) is then evi- /• dently a cross-cut of the original surface ; and CA is a cross-cut of the surface, which is the modification of the original surface after the loop-cut has been made. Since, by definition, a loop-cut does not meet the boundary, the cross-cut CA does not divide the modified surface into distinct pieces ; hence, according as the effect of the loop-cut is, \ Fi8- 41- or is not, that of making distinct pieces, so will the effect of the whole cross-cut be, or not be, that of making distinct pieces. 161. Let a loop-cut be drawn in a surface of connectivity N; as before for a cross-cut, there are two cases for consideration, according as the loop-cut does or does not divide the surface into distinct pieces. First, let it divide the surface into two distinct pieces, say of connectivities N! and N2 respectively. Change the loop-cut into a cross-cut of the original surface by drawing a cross-cut in either of the pieces, say the second, from a point in the course of the loop-cut to some point of the original boundary. This cross-cut, as a section of that piece, does not divide it into distinct pieces: and therefore the connectivity is now N? (= N2 — 1). The effect of the whole section, which is a single cross-cut, of the original surface is to divide it into two pieces, the connectivities of which are JVa and N2' : hence, by S 160, III., and therefore N1 + Na If the piece cut out be simply connected, say JVj. = 1, then the connectivity of the remainder is N + 1. But such a removal of a simply connected piece by a loop-cut is the same as making a hole in a continuous part of the surface : and therefore the effect of making a simple hole in a continuous part of a surface is to increase by unity the connectivity of the surface. If the piece cut out be doubly connected, say N: = 2, then the connect ivity of the remainder is N, the same as the connectivity of the original surface. Such a portion would be obtained by cutting out a piece with a hole in it which, so far as concerns the original surface, would be the same as merely enlarging the hole — an operation that naturally would not affect the connectivity. Secondly, let the loop -cut not divide the surface into two distinct pieces : and let N' be the connectivity of the modified surface. In this modified surface make a cross-cut k from any point of the loop-cut to a point of the boundary: this does not divide it into distinct pieces and therefore the connectivity after this last modification is N' -I. But the surface thus 161.] ON THE CONNECTIVITY 321 finally modified is derived from the original surface by the single cross-cut, constituted by the combination of k with the loop-cut : this single cross-cut does not divide the surface into distinct pieces and therefore the connectivity after the modification is N — 1. Hence that is, JV' = N, or the connectivity of a surface is not affected by a loop-cut which does not divide the surface into distinct pieces. Both of these results are included in the following theorem : — V. If after any number of loop-cuts made in a surface of connectivity N, there be r + 1 distinct pieces of surface, of connectivities JV^ JV2, ..., Nr+lt then N, + N3 + ...... + JVr+1 = JV+2r. Let the number of loop-cuts be s. Each of them can be changed into a cross-cut of the original surface, by drawing in some one of the pieces, as may be convenient, a cross-cut from a point of the loop-cut to a point of a boundary ; this new cross-cut does not divide the piece in which it is drawn into distinct pieces. If k such cross-cuts (where k may be zero) be drawn in the piece of connectivity Nm, the connectivity becomes Nm', where N ' — N~ — If- •" m — •*•• m I" j r+l r+l r+l hence 2 Nm' = 2 Nm-2k= X Nm - s. m=\ m-\ m=l We now have s cross-cuts dividing the surface of connectivity JV into r + l distinct pieces, of connectivities JV/, JV/, ..., JV/, Nr+1' ; and therefore, by § 160, IV., so that JVj + JV2 + . . . 4- Nr+1 = JV + 2r. This result could have been obtained also by combination and repetition of the two results obtained for a single loop-cut. Thus a spherical surface with one hole in it is simply connected : when n — l other different holes* are made in it, the edges of the holes being outside one another, the connectivity of the surface is increased by n— 1, that is, it becomes n. Hence a spherical surface with n holes in it is n-ply connected. 162. Occasionally, it is necessary to consider the effect of a slit made in the surface. If the slit have neither of its extremities on a boundary (and therefore no point on a boundary) it can be regarded as the limiting form of a loop-cut which makes a hole in the surface. Such a slit therefore (§ 161) increases the connectivity by unity. * These are holes in the surface, not holes bored through the volume of the sphere ; one of the latter would give two holes in the surface. F- 21 BOUNDARIES [162. If the slit have one extremity (but no other point) on a boundary, it can be regarded as the limiting form of a cross-cut, which returns on itself as in the figure, and cuts off a single simply con- / nected piece. Such a slit therefore (§ 160, III.) leaves the connectivity unaltered. If the slit have both extremities on boundaries, it ceases \ to be merely a slit : it is a cross-cut the effect of which on Fl8- 42- the connectivity has been obtained. We do not regard such sections as slits. 163. In the preceding investigations relative to cross-cuts and loop-cuts, reference has continually been made to the boundary of the surface con sidered. The boundary of a surface consists of a line returning to itself, or of a system of lines each returning to itself. Each part of such a boundary-line as it is drawn is considered a part of the boundary, and thus a boundary-line cannot cut itself and pass beyond its earlier position, for a boundary cannot be crossed: each boundary-line must therefore be a simple curve*. Most surfaces have boundaries : an exception arises in the case of closed surfaces whatever be their connectivity. It was stated (§ 159) that a boundary is assigned to such a surface by drawing an infinitesimal simple curve in it or, what is the same thing, by making a small hole. The advantage of this can be seen from the simple example of a spherical surface. When a small hole is made in any surface the connectivity is increased by unity : the connectivity of the spherical surface after the hole is made is unity, and therefore the connectivity of the complete spherical surface must be taken to be zero. The mere fact that the connectivity is less than unity, being that of the simplest connected surfaces with which we have to deal, is not in itself of importance. But let us return for a moment to the suggested method of determining the connectivity by means of the evanescence of circuits without crossing the boundary. When the surface is the complete spherical surface (Fig. 43), there are two essentially distinct ways of making a circuit C evan escent, first, by making it collapse into the point a, Fig. 43. secondly by making it expand over the equator and then collapse into the point b. One of the two is superfluous : it introduces an element of doubt as to the mode of evanescence unless that mode be specified a specification which in itself is tantamount to an assignment of * Also a line not returning to itself may be a boundary ; it can be regarded as the limit of a simple curve when the area becomes infinitesimal. 163.] EFFECT OF CROSS-CUTS ON BOUNDARIES 323 boundary. And in the case of multiply connected surfaces the absence of boundary, as above, leads to an artificial reduction of the connectivity by unity, arising not from the greater simplicity of the surface but from the possibility of carrying out in two ways the operation of reducing any circuit to given circuits, which is most effective when only one way is permissible. We shall therefore assume a boundary assigned to such closed surfaces as in the first instance are destitute of boundary. 164. The relations between the number of boundaries and the connect ivity of a surface are given by the following propositions. I. The boimdary of a simply connected surface consists of a single line. When a boundary consists of separate lines, then a cross-cut can be made from a point of one to a point of another. By proceeding from P, a point on one side of the cross-cut, along the boundary ac...cVwe can by a line lying wholly in the surface reach a point Q on the other side of the cross-cut : hence the parts of the surface on opposite sides of the cross-cut are connected. The surface is therefore not resolved into distinct pieces by the cross-cut. A simply connected surface is resolved into distinct pieces Fig. 44. by each cross-cut made in it : such a cross-cut as the foregoing is therefore not possible, that is, there are not separate lines which make up its boundary. It has a boundary : the boundary therefore consists of a single line. II. A cross-cut either increases by unity or diminishes by unity the number of distinct boundary -lines of a multiply connected surface. A cross-cut is made in one of three ways : either from a point a of one boundary-line A to a, point b of another boundary-line B ; or from a point a of a boundary-line to another point a' of the same boundary-line ; or from a point of a boundary-line to a point in the cut itself. If made in the first way, a combination of one edge of the cut, the remainder of the original boundary A, the other edge of the cut and the remainder of the original boundary B taken in succession, form a single piece of boundary ; this replaces the two boundary-lines A and B which existed distinct from one another before the cross-cut was made. Hence the number of lines is diminished by unity. An example is furnished by a plane ring (ii., Fig. 37, p. 314). If made in the second way, the combination of one edge of the cut with the piece of the boundary on one side of it makes one boundary-line, and the combination of the other edge of the cut with the other piece of the boundary makes another boundary-line. Two boundary-lines, after the cut is made, 21—2 324 NUMBER OF BOUNDARY-LINES [164. replace a single boundary-line, which existed before it was made : hence the number of lines is increased by unity. Examples are furnished by the cut surfaces in Fig. 40, p. 319. If made in the third way, the cross-cut may be considered as constituted by a loop-cut and a cut joining the loop-cut to the boundary. The boundary- lines may now be considered as constituted (Fig. 41, p. 320) by the closed curve ABD and the closed boundary abda'c'e'...eca; that is, there are now two boundary-lines instead of the single boundary-line ce...e'c'c in the uncut surface. Hence the number of distinct boundary-lines is increased by unity. COROLLARY. A loop-cut increases the number of distinct boundary-lines by two. This result follows at once from the last discussion. III. The number of distinct boundary-lines of a surface of connectivity N is N — 2k, where k is a positive integer that may be zero. Let m be the number of distinct boundary-lines ; and let N — 1 appro priate cross-cuts be drawn, changing the surface into a simply connected surface. Each of these cross-cuts increases by unity or diminishes by unity the number of boundary-lines ; let these units of increase or of decrease be denoted by e^ e2, ..., €#_!. Each of the quantities e is + 1 ; let k of them be positive, and N — 1 — k negative. The total number of boundary-lines is therefore m + k-(N-l-k). The surface now is a single simply connected surface, and there is therefore only one boundary-line ; hence m + k-(N-l-k) = l, so that m = N — 2k ; and evidently k is an integer that may be zero. COROLLARY 1. A closed surface with a single boundary-line* is of odd connectivity. For example, the surface of an anchor-ring, when bounded, is of con nectivity 3; the surface, obtained by boring two holes through the volume of a solid sphere, is, when bounded, of connectivity 5. If the connectivity of a closed surface with a single boundary be 2p + 1, the surface is often said-f- to be of class p (§ 178, p. 349.) COROLLARY 2. If the number of distinct boundary lines of a surface of connectivity N be N, any loop-cut divides the surface into two distinct pieces. After the loop-cut is made, the number of distinct boundary-lines is N+2; the connectivity of the whole of the cut surface is therefore not less * See § 159. t The German word is Geschlecht ; French writers use the word genre, and Italians genere. 164.] LHUILIER'S THEOREM 325 than N+2. It has been proved that a loop-cut, which does not divide the surface into distinct pieces, does not affect the connectivity ; hence as the connectivity has been increased, the loop-cut must divide the surface into two distinct pieces. It is easy, by the result of § 161, to see that, after the loop-cut is made, the sum of connectivities of the two pieces is N+2, so that the connectivity of the whole of the cut surface is equal to N + 2. Note. Throughout these propositions, a tacit assumption has been made, which is important for this particular proposition when the surface is the means of representing the variable. The assumption is that the surface is bifacial and not unifacial ; it has existed implicitly throughout all the geometrical representations of variability : it found explicit expression in § 4 when the plane was brought into relation with the sphere : and a cut in a surface has been counted a single cut, occurring in one face, though it would have to be counted as two cuts, one on each side, were the surface unifacial. The propositions are not necessarily valid, when applied to unifacial surfaces. Consider a surface made out of a long rectangular slip of paper, which is twisted once (or any odd number of times) and then has its ends fastened together. This surface is of double connectivity, because one section can be made across it which does not divide it into separate pieces ; it has only a single boundary-line, so that Prop. III. just proved does not ! apply. The surface is unifacial ; and it is possible, without meeting the boundary, to pass continuously in the surface from a point P to another point Q which could be reached merely by passing through the material at P. We therefore do not retain unifacial surfaces for consideration. 165. The following proposition, substantially due to Lhuilier*, may be taken in illustration of the general theory. If a closed surface of connectivity 2N + 1 (or of class N) be divided by circuits into any number of simply connected portions, each in the form of a curvilinear polygon, and if F be the number of polygons, E be the number of edges and S the number of angular points, then 2N=2 + JE-F-S. Let the edges E be arranged in systems, a system being such that any lino in it can be reached by passage along some other line or lines of the system ; let k be the number of such systems -f. To resolve the surface into a number of simply connected pieces composed of the F polygons, the cross-cuts will be made along the edges ; and therefore, unless a boundary be assigned * Gergonne, Ann. de Math., t. iii, (1813), pp. 181—186; see also Mobius, Ges. Werke, t. ii, p. 468. A circuit is defined in § 166. t The value of k is 1 for the proposition and is greater than 1 for the Corollary. 326 LHUILIER'S THEOREM [165. to the surface in each system of lines, the first cut for any system will be a loop-cut. We therefore take k points, one in each system as a boundary ; the first will be taken as the natural boundary of the surface, and the remaining k—\, being the limiting forms of k — 1 infinitesimal loop-cuts, increase the connectivity of the surface by k — 1, that is, the connectivity now is 2N+k. The result of the cross-cuts is to leave F simply connected pieces : hence Q, the number of cross-cuts, is given by At every angular point on the uncut surface, three or more polygons are contiguous. Let Sm be the number of angular points, where m polygons are contiguous; then Again, the number. of edges meeting at each of the S3 points is three, atl each of the $4 points is four, at each of the $5 points is five, and so on ; hence, in taking the sum 3$3 + 4$4 + 5$5 + . . ., each edge has been counted twice, once for each extremity. Therefore Consider the composition of the extremities of the cross-cuts ; the number of the extremities is 2Q, twice the number of cross-cuts. Each of the k points furnishes two extremities; for each such point is a boundary on which the initial cross-cut for each of the systems must begin and must end. These points therefore furnish 2k extremities. The remaining extremities occur in connection with the angular points. In making a cut, the direction passes from a boundary along an edge, past the point along another edge and so on, until a boundary is reached ; so that on the first occasion when a cross-cut passes through a point, it is made along two of the edges meeting at the point. Every other cross-cut passing through that point must begin or end there, so that each of the S3 points will furnish one extremity (corresponding to the remaining one cross-cut through the point), each of the $4 points will furnish two extremities (corresponding to the remaining two cross-cuts through the point), and so on. The total number of extremities thus provided is S3 + 2St+3S5 + ... Hence 2Q = 2k + 83 + 2St + 3S6+ ... or Q = k + E-S, which combined with Q = 2N + k + F - 2, leads to the relation 2N=2 + E-F-S. 165.] CIRCUITS ON CONNECTED SURFACES 327 The simplest case is that of a sphere, when Euler's relation F + S =• E + 2 is obtained. The case next in simplicity is that of an anchor-ring, for which the relation is F+ S = E. COROLLARY. If the result of making the cross-cuts along the various edges be to give the F polygons, not simply connected areas but areas of connectivities jYj + 1, jV2 + l, ..., Np+1 respectively, then the connectivity of the original surface is given by 166. The method of determining the connectivity of a surface by means of a system of cross-cuts, which resolve it into one or more simply connected pieces, will now be brought into relation with the other method, suggested in § 159, of determining the connectivity by means of irreducible circuits. A closed line drawn on the surface is called a circuit. A circuit, which can be reduced to a point by' continuous deformation without crossing the boundary, is called reducible ; a circuit, which cannot be so reduced, is called irreducible. An irreducible circuit is either (i) simple, when it cannot without crossing the boundary be deformed continuously into repetitions of one or more circuits ; or (ii) multiple, when it can without crossing the boundary be deformed continuously into repetitions of a single circuit ; or (iii) compound, when it can without crossing the boundary be deformed continuously into combinations of different circuits, that may be simple or multiple. The distinction between simple circuits and compound circuits, that involve no multiple circuits in their combination, depends upon conventions adopted for each particular case. A circuit is said to be reconcileable with the system of circuits into a combination of which it can be continuously deformed. If a system of circuits be reconcileable with a reducible circuit, the system is said to be reducible. As there are two directions, one positive and the other negative, in which a circuit can be described, and as there are possibilities of repetitions and of compositions of circuits, it is clear that circuits can be represented by linear algebraical expressions involving real quantities and having merely numerical coefficients. Thus a reducible circuit can be denoted by 0. If a simple irreducible circuit, positively described, be denoted by a, the same circuit, negatively described, can be denoted by — a. The multiple circuit, which is composed of m positive repetitions of the simple irreducible circuit a, would be denoted by ma ; but if the m repetitions were negative, the multiple circuit would be denoted by — ma. 328 CIRCUITS [106. A compound circuit, reconcileable with a system of simple irreducible circuits a1} a2, ..., an would be denoted by m1a1 + m2a2-\- ... + mnan, where mj, m2, ..., mn are positive or negative integers, being the net number of positive or negative descriptions of the respective simple irreducible circuits. The condition of the reducibility of a system of circuits al, «2, ..., an, each one of which is simple and irreducible, is that integers m1} m.2, ..., mn should exist such that m^j + m2a2 + . . . + mnan = 0, the sign of equality in this equation, as in other equations, implying that continuous deformation without crossing the boundary can change into one another the circuits, denoted by the symbols on either side of the sign. The representation of any compound circuit in terms of a system of independent irreducible circuits is unique : if there were two different expressions, they could be equated in the foregoing sense and this would imply the existence of a 'relation P& + p.2a2 + . . . +pnan = 0, which is excluded by the fact that the system is irreducible. Further, equations can be combined linearly, provided that the coefficients of the combinations be merely numerical. 167. In order, then, to be in a position to estimate circuits on a multiply connected surface, it is necessary that an irreducible system of irreducible simple circuits should be known, such a system being considered complete when every other circuit on the surface is reconcileable with the system. Such a system is not necessarily unique ; and it must be proved that, if more than one complete system be obtainable, any circuit can be reconciled with each system. First, the number of simple irreducible circuits in any complete system must be tlie same for the same surface. Let a1} ..., ap; and b1} ..., bn; be two complete systems. Because a1} ..., ap constitute a complete system, every circuit of the system of circuits b is reconcileable with it ; that is, integers ra# exist, such that br = mlral + m.2ra.2 + . . . + mprap, for r = 1, 2, ..., n. If n were >p, then by combining linearly each equation after the first p equations with those p equations, and eliminating al, ..., ap from the set of p + 1 equations, we could derive n —p relations of the form M^ + M,b2 + . . . + Mnbn = 0, where the coefficients M, being determinants the constituents of which are integers, would be integers. The system of circuits b is irreducible, and there are therefore no such relations ; hence n is not greater than p. 167.] ON CONNECTED SURFACES 329 Similarly, by considering the reconciliation of each circuit a with the irreducible system of circuits b, it follows that p is not greater than n. Hence p and n are equal to one another. And, because each system is a complete system, there are integers A and B such that ar = Arlbi + Ar2b.2 4- • • • + Arnbn (r = I, ..., bs = Bg^ + Bs.2a2 + . . . -I- BmOn (s = l, ..., The determinant of the integers A is equal to + 1 ; likewise the deter minant of the integers B. Secondly, let x be a circuit reconcileable with the system of circuits a : it is reconcileable with any other complete system of circuits. Since x is reconcileable with the system a, integers m1} ..., mn can be found such that x = ??i1«1 + . . . + mnan. Any other complete system of n circuits b is such that the circuits a can be expressed in the form ar = Anbj. + ... + Arnbn , (r = 1, . . ., n), where the coefficients A are integers ; and therefore n n n x = b1'2 mrArl 4- 62 S mrArz + . . . + bn X mrArn r=l r=l r=l = gri&i + gr2&a + ~'+qnl>n, where the coefficients q are integers, that is, x is reconcileable with the complete system of circuits b. 168. It thus appears that for the construction of any circuit on a surface, it is sufficient to know some one complete system of simple irreducible circuits. A complete system is supposed to contain the smallest possible number of simple circuits : any one which is reconcileable with the rest is omitted, so that the circuits of a system may be considered as independent. Such a system is indicated by the following theorems : — I. No irreducible simple circuit can be drawn on a simply connected surface*. If possible, let an irreducible circuit G be drawn in a simply connected surface with a boundary B. Make a loop-cut along C, and change it into a cross-cut by making a cross-cut A from some point of C to a point of B ; this cross-cut divides the surface into two simply connected pieces, one of which is bounded by B, the two edges of A, and one edge of the cut along C, and the other of which is bounded entirely by the cut along C. The latter surface is smaller than the original surface ; it is simply connected and has a single boundary. If an irreducible simple circuit can be drawn on it, we proceed as before, and again obtain a still smaller simply connected surface. In this way, we ultimately obtain an infinitesimal * All surfaces considered are supposed to be bounded. 330 RELATIONS BETWEEN CONNECTIVITY [168. element ; for every cut divides the surface, in which it is made, into distinct pieces. Irreducible circuits cannot be drawn in this element ; and therefore its boundary is reducible. This boundary is a circuit in a larger portion of the surface : the circuit is reducible so that, in that larger portion no irreducible circuit is possible and therefore its boundary is reducible. This boundary is a circuit in a still larger portion, and the circuit is reducible : so that in this still larger portion no irreducible circuit is possible and once more the boundary is reducible. Proceeding in this way, we find that no irreducible simple circuit is possible in the original surface. COROLLARY. No irreducible circuit can be drawn on a simply connected surface. II. A complete system of irreducible simple circuits for a surface of connectivity N contains N— I simple circuits, so that every other circuit on the surface is reconcileable with that system. Let the surface be resolved by cross-cuts into a single simply connected surface: N— 1 cross-cuts will be necessary. Let CD be any one of them : and let a and b be two points on the /e opposite edges of the cross-cut. Then since the surface is L n simply connected, a line can be drawn in the surface from a to b without passing out of the surface or without meeting a part of the boundary, that is, without meeting any other cross-cut. The cross-cut CD ends either in Fis- 45- another cross-cut or in a boundary; the line ae...fb surrounds that other cross-cut or that boundary as the case may be : hence, if the cut CD be obliterated, the line ae...fba is irreducible on the surface in which the other N — 2 cross-cuts are made. But it meets none of those cross cuts; hence, when they are all obliterated so as to restore the unresolved surface of connectivity N, it is an irreducible circuit. It is evidently riot a repeated circuit; hence it is an irreducible simple circuit. Hence the line of an irreducible simple circuit on an unresolved surface is given by a line passing from a point on one edge of a cross-cut in the resolved surface to a point on the opposite edge. Since there are N -I cross-cuts, it follows that N —1 irreducible simple circuits can thus be obtained: one being derived in the foregoing manner from each of the cross-cuts, which are necessary to render the surface simply connected. It is easy to see that each of the irreducible circuits on an unresolved surface is, by the cross-cuts, rendered impossible as a circuit on the resolved surface. But every other irreducible circuit C is reconcileable with the N—l circuits, thus obtained. If there be one not reconcileable with these N-l circuits, then, when all the cross-cuts are made, the circuit C is not rendered 168.] AND IRREDUCIBLE CIRCUITS 331 impossible, if it be not reconcileable with those which are rendered impossible by the cross-cuts : that is, there is on the resolved surface an irreducible circuit. But the resolved surface is simply connected, and therefore no irreducible circuit can be drawn on it : hence the hypothesis as to C, which leads to this result, is not tenable. Thus every other circuit is reconcileable with the system of N — 1 circuits : and therefore the system is complete*. This method of derivation of the circuits at once indicates how far a system is arbitrary. Each system of cross-cuts leads to a complete system of irreducible simple circuits, and vice versa ; as the one system is not unique, so the other system is not unique. For the general question, Jordan's memoir, Des contours traces sur les surfaces, Liouville, 2me Ser., t. xi., (1866), pp. 110—130, may be consulted. Ex. 1. On a doubly connected surface, one irreducible simple circuit can be drawn. It is easily obtained by first resolving the surface into one that is simply connected — a single cross-cut CD is effective for this purpose — and then by drawing a curve aeb in the Fig. 46, (i). surface from one edge of the cross-cut to the other. All other irreducible circuits on the unresolved surface are reconcileable with the circuit aeba. Ex. 2. On a triply- connected surface, two independent irreducible circuits can be Fig. 46, (ii). * If the number of independent irreducible simple circuits be adopted as a basis for the definition of the connectivity of a surface, the result of the proposition would be taken as the definition : and the resolution of the surface into one, which is simply connected, would then be obtained by developing the preceding theory in the reverse order. 332 DEFORMATION [168. drawn. Thus in the figure Cl and C2 will form a complete system. The circuits C3 and (74 are also irreducible : they can evidently be deformed into C^ and <72 and reducible circuits by continuous deformation : in the algebraical notation adopted, we have C3=C1 + C2, Ci=Cl-C.2. Ex. 3. Another example of a triply connected surface is given in Fig. 47. Two irredu cible simple circuits are Cv and C%. Another irreducible circuit is C3; this can be Fig. 47. reconciled with Cl and C.2 by drawing the point a into coincidence with the intersection of Cj and (72, and the point c into coincidence with the same point. Ex. 4. As a last example, consider the surface of a solid sphere with n holes bored through it. The connectivity is 2n + 1 : hence 2n independent irreducible simple circuits Fig. 48. can be drawn on the surface. The simplest complete system is obtained by taking 2n curves : made up of a set of n, each round one hole, and another set of n, each through one hole. A resolution of this surface is given by taking cross-cuts, one round each hole (making the circuits through the holes no longer possible) and one through each hole (making the circuits round the holes no longer possible). The simplest case is that for which n= 1 : the surface is equivalent to the anchor-ring. 169. Surfaces are at present being considered in view of their use as a means of representing the value of a complex variable. The foregoing inves tigations imply that surfaces can be classed according to their connectivity ; and thus, having regard to their designed use, the question arises as to whether all surfaces of the same connectivity arc equivalent to one another, so as to be transformable into one another. 169.] OF CONNECTED SURFACES 333 Moreover, a surface can be physically deformed and still remain suitable for representation of the variable, provided certain conditions are satisfied. We thus consider geometrical transformation as well as physical deformation ; but we are dealing only with the general results and not with the mathematical relations of stretching and bending, which are discussed in treatises on Analytical Geometry*. It is evident that continuity is necessary for both : discontinuity would imply discontinuity in the representation of the variable. Points that are contiguous (that is, separated only by small distances measured in the surface) must remain contiguous -f*: and one point in the unchanged surface must correspond to only one point in the changed surface. Hence in the continuous deformation of a surface there may be stretching and there may be bending ; but there must be no tearing and there must be no joining. For instance, a single untwisted ribbon, if cut, comes to be simply connected. If a twist through 180° be then given to one end and that end be then joined to the other, we shall have a once- twisted ribbon, which is a surface with only one face and only one edge; it cannot be looked upon as an equivalent of the former surface. A spherical surface with a single hole can have the hole stretched and the surface flattened, so as to be the same as a bounded portion of a plane : the two surfaces are equivalent to one another. Again, in the spherical surface, let a large indentation be made : let both the outer and the inner surfaces be made spherical ; and let the mouth of the indentation be contracted into the form of a long, narrow hole along a part of a great circle. When each point of the inner surface is geometrically moved so that it occupies the position of its reflexion in the diametral plane of the hole, the final form§ of the whole surface is that of a two-sheeted surface with a junction along a line : it is a spherical winding-surface, and is equivalent to the simply connected spherical surface. 170. It is sufficient, for the purpose of representation, that the two surfaces should have a point-to-point transformation : it is not necessary that physical deformation, without tears or joins, should be actually possible. Thus a ribbon with an even number of twists would be as effective as a limited portion of a cylinder, or (what is the same thing) an untwisted ribbon : but it is not possible to deform the one into the other physically |. It is easy to see that either deformation or transformation of the kind considered will change a bifacial surface into a bifacial surface ; that it will not alter the connectivity, for it will not change irreducible circuits into * See, for instance, Frost's Solid Geometry, (3rd ed.), pp. 342 — 352. t Distances between points must be measured along the surface, not through space ; the distance between two points is a length which one point would traverse before reaching the position of the other, the motion of the point being restricted to take place in the surface. Examples will arise later, in Biemann's surfaces, in which points that are contiguous in space are separated by finite distances on the surface. § Clifford, Coll. Hath. Papers, p. 250. J The difference between the two cases is that, in physical deformation, the surfaces are the surfaces of continuous matter and are impenetrable ; while, in geometrical transformation, the surfaces may be regarded as penetrable without interference with the continuity. 334 DEFORMATION OF SURFACES [170. reducible circuits, and the number of independent irreducible circuits determines the connectivity: and that it will not alter the number of boundary curves, for a boundary will be changed into a boundary. These are necessary relations between the two forms of the surface : it is not difficult to see that they are sufficient for correspondence. For if, on each of two bifacial surfaces with the same number of boundaries and of the same connectivity, a complete system of simple irreducible circuits be drawn, then, when the members of the systems are made to correspond in pairs, the full transformation can be effected by continuous deformation of those corresponding irreducible circuits. It therefore follows that : — The necessary and sufficient conditions, that two bifacial surfaces may be equivalent to one another for the representation of a variable, are that tlie two surfaces should be of the same connectivity and should have the same number of boundaries. As already indicated, this equivalence is a geometrical equivalence : deformation may be (but is not of necessity) physically possible. Similarly, the presence of one or of several knots in a surface makes no essential difference in the use of the surface for representing a variable. Thus a long cylindrical surface is changed into an anchor-ring when its ends are joined together ; but the changed surface would be equally effective for purposes of representation if a knot were tied in the cylindrical surface before the ends are joined. But it need hardly be pointed out that though surfaces, thus twisted or knotted, are equivalent for the purpose indicated, they are not equivalent for all topological enumerations. Seeing that bifacial surfaces, with the same connectivity and the same number of boundaries, are equivalent to one another, it is natural to adopt, as the surface of reference, some simple surface with those characteristics; thus for a surface of connectivity 2p + 1 with a single boundary, the surface of a solid sphere, bounded by a point and pierced through with p holes, could be adopted. Klein calls* such a surface of reference a Normal Surface. It has been seen that a bounded spherical surface and a bounded simply connected part of a plane are equivalent — they are, moreover, physically deformable into one another. An untwisted closed ribbon is equivalent to a bounded piece of a plane with one hole in it — they are deformable into one another : but if the ribbon, previous to being closed, have undergone an even number of twists each through 180°, they are still equivalent but are not physically deformable into one another. Each of the bifacial surfaces is doubly connected (for a single cross-cut renders each simply connected) and each of them * Ueber Riemann's Theorie der algebraischen Functionen und ihrer Integrate, (Leipzig, Teubner, 1882), p. 26. 170.] REFERENCES 335 has two boundaries. If however the ribbon, previous to being closed, have imdcrgone an odd number of twists each through 180°, the surface thus obtained is not equivalent to the single-holed portion of the plane ; it is unifacial arid has only one boundary. A spherical surface pierced in n-\-l holes is equivalent to a bounded portion of the plane with n holes ; each is of connectivity n + 1 and has n + 1 boundaries. The spherical surface can be deformed into the plane surface by stretching one of its holes into the form of the outside boundary of the plane surface. Ex. Prove that the surface of a bounded anchor-ring can be physically deformed into the surface in Fig. 47, p. 332. For continuation and fuller development of the subjects of the present chapter, the following references, in addition to those which have been given, will be found useful : Klein, Math. Ann., t. vii, (1874), pp. 548—557; ib., t. ix, (1876), pp. 476—482. Lippich, Math. Ann., i. vii, (1874), pp. 212 — 229 ; Wiener Sitzungsb., t. Ixix, (ii), (1874), pp. 91—99. Durege, Wiener Sitzungsb., t. Ixix, (ii), (1874), pp. 115—120; and section 9 of his treatise, quoted on p. 316, note. Neumann, chapter vii of his treatise, quoted on p. 5, note. Dyck, Math. Ann., t. xxxii, (1888), pp. 457—512, ib., t. xxxvii, (1890), pp. 273—316; at the beginning of the first part of this investigation, a valuable series of references is given. Dingeldey, Topologische Studien, (Leipzig, Teubner, 1890). CHAPTER XV. RIEMANN'S SURFACES. 171. THE method of representing a variable by assigning to it a position in a plane or on a sphere is effective when properties of uniform functions of that variable are discussed. But when multiform functions, or integrals of uniform functions occur, the method is effective only when certain parts of the plane are excluded, due account being subsequently taken of the effect of such exclusions; and this process, the extension of Cauchy's method, was adopted in Chapter IX. There is another method, referred to in § 100 as due to Riemann, of an entirely different character. In Riemann's representation, the region, in which the variable z exists, no longer consists of a single plane but of a number of planes ; they are distinct from one another in geometrical concep tion, yet, in order to preserve a representation in which the value of the variable is obvious on inspection, the planes are infinitesimally close to one another. The number of planes, often called sheets, is the same as the number of distinct values (or branches) of the function w for a general argument z and, unless otherwise stated, will be assumed finite; each sheet is associated with one branch of the function, and changes from one branch of the function to another are effected by making the ^-variable change from one sheet to another, so that, to secure the possibility of change of sheet, it is necessary to have means of passage from one sheet to another. The aggregate of all the sheets is a surface, often called a Riemanns Surface. For example, consider the function w=z* + (z-I}~*, the cube roots being independent of one another. It is evidently a nine-valued function ; the number of sheets in the appropriate Eiemann's surface is therefore nine. The branch-points are 2 = 0, z = l, 2=00. Let o> and a denote a cube-root of unity, independently of one another ; then the values of z* can be represented in the form 171.] EXAMPLES OF RIEMANN's SURFACES 337 ill -A - 4 23, C023", co22*; and the values of (2-!) 3 can be represented in the form (2-!) , ^•(z - \ ) ~ 3} 0 (« - 1) » The nine values of w can be symbolically expressed as follows : — Fig. 49. Fig. 50. where the symbols opposite to w give the coefficients of z3 and of (2- 1) 3 respectively. Now when 2 describes a small simple circuit positively round the origin, the groups in cyclical order are u\, w2, w3; w4, w5, w6; wr, w8, io9. And therefore, in the immediate vicinity of the origin, there must be means of passage to enable the 2-point to make the corresponding changes from sheet to — sheet. Taking a section of the whole surface near the origin ~ so as to indicate the passages and regarding the right-hand sides as the part from which the 2-variable moves when it — describes a circuit positively, the passages must be in character as indicated in Fig. 49. And it is evident that the further descrip tion of small simple circuits round the origin will, with these passages, lead to the proper values : thus %, which after the single description is the value of w4, becomes w6 after another description and it is evident that a point in the w-0 sheet passes into the w6 sheet. When 2 describes a small simple circuit positively round the point 1, the groups in cyclical order are wlt ^4, %; w2, w5, ws; w3, w6, w9: and therefore, in the immediate vicinity of the point 1, there must be ~ means of passage to render possible the corresponding changes of 2 from sheet to sheet. Taking a section as before near the ~ point 1 and with similar convention as to the positive direc tion of the 2-path, the passages must be in character as indicated in Fig. 50. Similarly for infinitely large values of 2. If then the sheets can be so joined as to give these possibilities of passage and also give combinations of them corresponding to combinations of the simple paths indicated, then there will be a surface to any point of which will correspond one and only one value of w : and when the value of w is given for a point 2 in an ordinary plane of variation, then that value of w will determine the sheet of the surface in which the point 2 is to be taken. A surface will then have been constructed such that the function w, which is multiform for the single-plane representation of the variable, is uniform for variations in the many-sheeted surface. Again, for the simple example arising from the two-valued function, defined by the equation w = {(z-a}(z-b}(z-c}}-\ the branch-points are a, b, c, oo ; and a small simple circuit round any one of these four points interchanges the two values. The Riemann's surface is two-sheeted and there must be means of passage between the two sheets in the vicinity of a, that of b, that of c and at the infinite part of the plane. These examples are sufficient to indicate the main problem. It is the construction of a surface in which the independent variable can move so F. 22 338 SHEETS OF HIEMANN'S SURFACE [171. that, for variations of z in that surface, the multiformity of the function is changed to uniformity. From the nature of the case, the character of the surface will depend on the character of the function : and thus, though all the functions are uniform within their appropriate surfaces, these surfaces are widely various. Evidently for uniform functions of z the appropriate surface on the above method is the single plane already adopted. 172. The simplest classes of functions for which a Riemaim's surface is useful are (i) those called (§ 94) algebraic functions, that is, multiform functions of the independent variable denned by an algebraical equation of the form which is of finite degree, say n, in w, and (ii) those usually called Abelian functions, which arise through integrals connected with algebraic functions. Of such an algebraic function there are, in general, n distinct values ; but for the special values of z, that are the branch-points, two or more of the values coincide. The appropriate Riemann's surface is composed of n sheets ; one branch, and only one branch, of w is associated with a sheet. The variable z, in its relation to the function, is determined not merely by its modulus and argument but also by its sheet ; that is, in the language of the earlier method, we take account of the path by which z acquires a value. The particular sheet in which z lies determines the particular branch of the function. Variations of #, which occur within a sheet and do not coincide with points lying in regions of passage between the sheets, lead to variations in the value of the branch of w associated with the sheet ; a return to an initial value of z, by a path that nowhere lies within a region of passage, leaves the ^-point in the same sheet as at first and so leads to the initial branch (and to the initial value of the branch) of w. But a return to an initial value of z by a path, which, in the former method of representation, would enclose a branch-point, implies a change of the branch of the function according to the definite order prescribed by the branch-point. Hence the final value of the variable z on the Riemann's surface must lie in a sheet that is different from that of the initial (and algebraically equal) value ; and therefore the sheets must be so connected that, in the immediate vicinity of branch-points, there are means of passage from one sheet to another, securing the proper interchanges of the branches of the function as defined by the equation. 173. The first necessity is therefore the consideration of the mode in which the sheets of a Riemann's surface are joined : the mode is indicated by the theorem that sheets of a Riemann's surface are joined along lines. The junction might be made either at a point, as with two spheres in contact, or by a common portion of a surface, as with one prism lying on 173.] JOINED ALONG BRANCH-LINES 339 another, or along lines ; but whatever the character of the junction be, it must be such that a single passage across it (thereby implying entrance to the junction and exit from it) must change the sheet of the variable. If the junction were at a point, then the £- variable could change from one sheet into another sheet, only if its path passed through that point : any other closed path would leave the z- variable in its original sheet. A small closed curve, infinites! rn ally near the point and enclosing it and no other branch-point, is one which ought to transfer the variable to another sheet because it encloses a branch-point : and this is impossible with a point-junction when the path does not pass through the point. Hence a junction at a point only is insufficient to provide the proper means of passage from sheet to sheet. If the junction were effected by a common portion of surface, then a passage through it (implying an entrance into that portion and an exit from it) ought to change the sheet. But, in such a case, closed contours .-'--'' can be constructed which make such a passage without Fi8- 51> enclosing the branch-point a : thus the junction would cause a change of sheet for certain circuits the description of which ought to leave the z- variable in the original sheet. Hence a junction by a continuous area of surface does not provide proper means of passage from sheet to sheet. The only possible junction which remains is a line. The objection in the last case does not apply to a closed • / '^ contour which does not contain the branch-point ; for the /.--"'' line cuts the curve twice and there are therefore two Fig. 52. crossings ; the second of them makes the variable return to the sheet which the first crossing compelled it to leave. Hence the junction between any two sheets takes place along a line. Such a line is called* a branch-line. The branch -points of a multiform function lie on the branch-lines, after the foregoing explanations ; and a branch-line can be crossed by the variable only if the variable change its sheet at crossing, in the sequence prescribed by the branch-point of the function which lies on the line. Also, the sequence is reversed when the branch-line is crossed in the reversed direction. Thus, if two sheets of a surface be connected along a branch-line, a point which crosses the line from the first sheet must pass into the second and a point which crosses the line from the second sheet must pass into the first. Again, if, along a common direction of branch-line, the first sheet of a surface be connected with the second, the second with the third, and the third with * Sometimes cross-line, sometimes branch-section. The German title is Verzweigungschnitt; the French is lignc de passage ; see also the note on the equivalents of branch-point, p. 15. 22—2 340 PROPERTIES OF BRANCH-LINES [173. the first, a point which crosses the line from the first sheet in one direction must pass into the second sheet, but if it cross the line in the other direction it must pass into the third sheet. A branch -point does not necessarily affect all the branches of a function : when it affects only some of them, the corresponding property of the Riemann's surface is in evidence as follows. Let z=a determine a branch-point affecting, say, only r branches. Take n points a, one in each of the sheets ; and through them draw n lines cab, having the same geometrical position in the respective sheets. Then in the vicinity of the point a in each of the n sheets, associated with the r affected branches, there must be means of passage from each one to all the rest of them ; and the lines cab can conceivably be the branch-lines with a properly established sequence. The point a does not affect the other n — r branches : there is therefore no necessity for means of passage in the vicinity of a among the remaining n — r sheets. In each of these remaining sheets, the point a and the line cab belong to their respective sheets alone : for them, the point a is not a branch-point and the line cab is not a branch- line. 174. Several essential properties of the branch-lines are immediate inferences from these conditions. I. A free end of a branch-line in a surface is a branch-point. Let a simple circuit be drawn round the free end so small as to enclose no branch-point (except the free end, if it be a branch-point). The circuit meets the branch-line once, and the sheet is changed because the branch-line is crossed ; hence the circuit includes a branch-point which therefore can be only the free end of the line. Note. A branch-line may terminate in the boundary of the surface, and then the extremity need not be a branch-point. II. When a branch-line extends beyond a branch-point lying in its course, the sequence of interchange is not the same on the two sides of the point. If the sequence of interchange be the same on the two sides of the branch point, a small circuit round the point would first cross one part of the branch- line and therefore involve a change of sheet and then, in its course, would cross the other part of the branch-line in the other direction which, on the supposition of unaltered sequence, would cause a return to the initial sheet. In that case, a circuit round the branch-point would fail to secure the proper change of sheet. Hence the sequence of interchange caused by the branch- line cannot be the same on the two sides of the point. III. If two branch-lines with different sequences of interchange have a common extremity, that point is either a branch-point or an extremity of at least one other branch-line. 174.] SYSTEM OF BRANCH-LINES 341 If the point be not a branch-point, then a simple curve enclosing it, taken so small as to include no branch-point, must leave the variable in its initial sheet. Let A be such a point, AB and AC be two branch- lines having A for a common extremity ; let ., A ,.• — ^ « the sequence be as in the figure, taken for a F. simple case ; and suppose that the variable initially is in the rth sheet. A passage across AB makes the variable pass into the sth sheet. If there be no branch-line between AB and AC having an extremity at A, and if neither n nor m be s, then the passage across AC makes no change in the sheet of the variable and, therefore, in order to restore r before AB, at least one branch-line must lie in the angle between AC and AB, estimated in the positive trigonometrical sense. If either n or m, say n, be s, then after passage across AC, the point is in the mt\i sheet ; then, since the sequences are not the same, m is not r and there must be some branch-line between AC and AB to make the point return to the rth sheet on the completion of the circuit. If then the point A be not a branch-point, there must be at least one other branch-line having its extremity at A. This proves the proposition. COROLLARY 1. If both of two branch-lines extend beyond a point of inter section, which is not a branch-point, no sheet of the surface has both of them for branch-lines. COROLLARY 2. If a change of sequence occur at any point of a branch- line, then either that point is a branch-point or it lies also on some other branch-line. COROLLARY 3. No part of a branch-line with only one branch-point on it can be a closed curve. It is evidently superfluous to have a branch-line without any branch-point on it. 175. On the basis of these properties, we can obtain a system of branch- lines satisfying the requisite conditions which are : — (i) the proper sequences of change from sheet to sheet must be secured by a description of a simple circuit round a branch point : if this be satisfied for each of the branch-points, it will evidently be satisfied for any combination of simple circuits, that is, for any path whatever enclosing one or more branch points. (ii) the sheet, in which the variable re-assumes its initial value after describing a circuit that encloses no branch-point, must be the initial sheet. 342 SYSTEM OF BRANCH-LINES [175. In the ^-plane of Cauchy's method, let lines be drawn from any point I, not a branch-point in the first instance, to each of the branch-points, as in fig. 19, p. 156, so that the joining lines do not meet except at /: and suppose the w-sheeted Riemann's surface to have branch-lines coinciding geometrically with these lines, as in § 173, and having the sequence of interchange for passage across each the same as the order in the cycle of functional values for a small circuit round the branch-point at its free end. No line (or part of a line) can be a closed curve ; the lines need not be straight, but they will be supposed drawn as direct as possible to the points in angular succession. The first of the above requisite conditions is satisfied by the establish ment of the sequence of interchange. To consider the second of the conditions, it is convenient to divide circuits into two kinds, (a) those which exclude /, (/3) those which include /, no one of either kind (for our present purpose) including a branch-point. A closed circuit, excluding I and all the branch-points, must intersect a branch-line an even number of times, if it intersect the line in real points. Let the figure (fig. 54) represent such a case : then the crossings at A and B counter act one another and so the part be tween A and B may without effect be transferred across IB3 so as not to cut the branch-line at all. Similarly for the points C and D : and a similar transference of the part now between C and D may be made across the branch-line without effect: that is, the circuit can, without effect, be changed so as not to cut the branch-line IBS at all. A similar change can be made for each of the branch-lines : and so the circuit can, without effect, be changed into one which meets no branch-line and therefore, on its completion, leaves the sheet unchanged. A closed circuit, including / but no branch-point, must meet each branch- line an odd number of times. A change similar in character to that in the previous case may be made for each branch-line : and without affecting the result, the circuit can be changed so that it meets each branch-line only once. Now the effect produced by a branch-line on the function is the same as the description of a simple loop round the branch-point which with / determines the branch-line : and therefore the effect of the circuit at present contemplated is, after the transformation which does not affect the result, the same as that of a circuit, in the previously adopted mode of representation, 175.] FOR A SURFACE enclosing all the branch-points. But, by Cor. III. of § 90, the effect of a circuit which encloses all the branch-points (including z = GO , if it be a branch-point) is to restore the value of the function which it had at the beginning of the circuit : and therefore in the present case the effect is to make the point return to the sheet in which it lay initially. It follows therefore that, for both kinds of a closed circuit containing no branch-point, the effect is to make the ^-variable return to its initial sheet on resuming its initial value at the close of the circuit. Next, let the point / be a branch-point ; and let it be joined by lines, as direct* as possible, to each of the other branch -points in angular succes sion. These lines will be regarded as the branch-lines ; and the sequence of interchange for passage across any one is made that of the interchange pre scribed by the branch-point at its free extremity. The proper sequence of change is secured for a description of a simple closed circuit round each of the branch-points other than /. Let a small circuit be described round /; it meets each of the branch-lines once and therefore its effect is the same as, in the language of the earlier method of representing variation of z, that of a circuit enclosing all the branch-points except 7. Such a circuit, when taken on the Neumann's sphere, as in Cor. III., § 90 and Ex. 2, § 104, may be regarded in two ways, according as one or other of the portions, into which it divides the area of the sphere, is regarded as the included area; in one way, it is a circuit enclosing all the branch points except /, in the other it is a circuit enclosing / alone and no other branch-point. Without making any modification in the final value of w, it can (by § 90) be deformed, either into a succession of loops round all the branch-points save one, or into a loop round that one ; the effect of these two deformations is therefore the same. Hence the effect of the small closed circuit round / meeting all the branch-lines is the same as, in the other mode of representation, that of a small curve round / enclosing no other branch point ; and therefore the adopted set of branch- lines secures the proper sequence of change of value for description of a circuit round /. The first of the two necessary conditions is therefore satisfied by the present arrangement of branch-lines. The proof, that the second of the two necessary conditions is also satisfied by the present arrangement of branch-lines, is similar to that in the preceding case, save that only the first kind of circuit of the earlier proof is possible. Jt thus appears that a system of branch-lines can be obtained which secures the proper changes of sheet for a multiform function : and therefore Riemann's surfaces can be constructed for such a function, the essential property being that over its appropriate surface an otherwise multiform function of the variable is a uniform function. * The reason for this will appear in §§ 183, 184. 344 EXAMPLES [175. The multipartite character of the function has its influence preserved by the character of the surface to which the function is referred : the surface, consisting of a number of sheets joined to one another, may be a multiply connected surface. In thus proving the general existence of appropriate surfaces, there has remained a large arbitrary element in their actual construction : moreover, in particular cases, there are methods of obtaining varied configurations of branch-lines. Thus the assignment of the n branches to the n sheets has been left unspecified, and is therefore so far arbitrary : the point I, if not a branch-point, is arbitrarily chosen and so there is a certain arbitrariness of position in the branch -lines. Naturally, what is desired is the simplest appropriate surface : the particularisation of the preceding arbitrary qualities is used to derive a canonical form of the surface. 176. The discussion of one or two simple cases will help to illustrate the mode of junction between the sheets, made by branch-lines. The simplest case of all is that in which the surface has only a single sheet: it does not require discussion. The case next in simplicity is that in which the surface is two-sheeted : the function is therefore two- valued and is consequently defined by a quadratic equation of the form Lua + 2Mu + N = 0, where L and M are uniform functions of z. When a new variable w is introduced, defined by Lu + M=w, so that values of iv and of u correspond uniquely, the equation is It is evident that every branch-point of u is a branch-point of w, and vice versa ; hence the Riemann's surface is the same for the two equations. Now any root of P (z) of odd degree is a branch-point of iv. If then where R (z} is a product of only simple factors, every factor of R (z) leads to a branch-point. If the degree of R (z} be even, the number of branch-points for finite values of the variable is even and z = oo is not a branch-point ; if the degree of R(z) be odd, the number of branch -points for finite values of the variable is odd and z = oo is a branch-point : in either case, the number of branch-points is even. There are only two values of w, and the Riemann's surface is two-sheeted: crossing a branch-line therefore merely causes a change of sheet. The free ends of branch-lines are branch-points ; a small circuit round any branch point causes an interchange of the branches w, and a circuit round any two branch-points restores the initial value of w at the end and therefore leaves the variable in the same sheet as at the beginning. These are the essential requirements in the present case ; all of them are satisfied by taking each 176.] OF RIEMANN'S SURFACES 345 branch-line as a line connecting two (and only two) of the branch-points. The ends of all the branch -lines are free : and their number, in this method, is one-half that of the (even) number of branch-points. A small circuit round a branch-point meets a branch-line once and causes a change of sheet ; a circuit round two (and not more than two) branch -points causes either no crossing of branch-line or an even number of crossings and therefore restores the variable to the initial sheet. A branch-line is, in this case, usually drawn in the form of a straight line when the surface is plane : but this form is not essential and all that is desirable is to prevent intersections of the branch-lines. Note. Junction between the sheets along a branch-line is easily secured. The two sheets to be joined are cut along the branch-line. One edge of the cut in the upper sheet, say its right edge looking along the section, is joined to the left edge of the cut in the lower sheet ; and the left edge in the upper sheet is joined to the right edge in the lower. A few simple examples will illustrate these remarks as to the sheets : illustrations of closed circuits will arise later, in the consideration of integrals of multiform functions. Ex. 1. Let w* = A(z-a)(z-b}, so that a and b are the only branch-points. The surface is two-sheeted : the line ab may be made the branch-line. In Fig. 55 only part of the upper sheet is shewn*, as likewise only part of the lower sheet. Continuous lines imply what is visible ; arid dotted lines what is invisible, on the supposition that the sheets are opaque. The circuit, closed in the surface and passing round 0, is made up of OJK in the upper sheet : the point crosses the branch-line and then passes into the lower sheet, where it describes the dotted line KLH : it then meets and crosses the branch-line at If, passes into the upper sheet and in that sheet returns to 0. Similarly of the line ABC, the part AB lies in the lower sheet, the part EC in the upper : of the line DG the part DE lies in the upper sheet, the part EFG in the lower, the piece FG of this part being there visible beyond the boundary of the retained portion of the upper surface. Ex. 2. Let Aw?2 = z3-a3. The branch-points (Fig. 56) are A ( = a), B ( = ««), (7( = aa2), where a is a primitive cube root of unity, and 2 = 00. The branch -lines can be made by BC, Ace ; and the two- sheeted surface will be a surface over which w is uniform. Only a part of each sheet is shewn in the figure; a section also is made at M across the surface, cutting the branch - line A QO . Ex. 3. Let wm=zn, where n and TO are prime to each other. The branch-points are z = 0 and 2=00 ; and the branch-line extends from 0 to QO . There are m sheets ; if we associate them in order with the branches ws, where wa=re for s=l, 2, ..., TO, then the first sheet is connected with the second forwards, the second with the third forwards, and so on ; the mth being connected with the first forwards. * The form of the three figures in the plate opposite p. 346 is suggested by Holzmiiller, Ein- fiihrung in die Theorie der isogonalen Vericandschaften und der confomien AbbUdimgen, (Leipzig, Teubner, 1882), in which several illustrations are given. 346 SPHERICAL RIEMANN'S SURFACE [176. The surface is sometimes also called a winding-surface; and a branch-point such as z—0 on the surface, where a number m of sheets pass into one another in succession, is also called a winding-point of order m— 1 (see p. 15, note). An illustration of the surface for m = 3 is given in Fig. 57, the branch-line being cut so as to shew the branching : what is visible is indicated by continuous lines ; what is in the second sheet, but is invisible, is indicated by the thickly dotted line ; what is in the third sheet, but is invisible, is indic ated by the thinly dotted line. Ex. 4. Consider a three-sheeted surface having four branch-points at a, b, c, d ; and let each point interchange two branches, say, w.2, w3 at a ; iv^ w3ai b ; w2, w3 at c ; wlt w2 at d ; the points being as in Fig. 58. It is easy to verify that these branch-points satisfy the condition that a circuit, enclosing them all, restores the initial value of w. The branching of the sheets may be made as in the figure, the integers on the two sides of the line indicating the sheets that are to be joined along the line. A canonical form for such a surface can be derived from the more general case given later (in §§ 186—189). Ex. 5. Shew that, if the equation be of degree n in w and be irreducible, all the n sheets of the surface are connected, that is, it is possible by an appropriate path to pass from any sheet to any other sheet. 177. It is not necessary to limit the surface representing the variable to a set of planes; and, indeed, as with uniform functions, there is a convenience in using the sphere for the purpose. We take n spheres, each of diameter unity, touching the Riemann's plane surface at a point A ; each sphere is regarded as the stereographic projection of a plane sheet, with regard to the other extremity A' of the spherical diameter through A. Then, the sequence of these spherical sheets being the same as the sequence of the plane sheets, branch-points in the plane surface project into branch-points on the spherical surface : branch -lines be tween the plane sheets project into branch-lines between the spherical sheets and are terminated by corresponding points ; and if a branch-line extend in the plane surface to z=co, the corresponding branch-line in the spherical surface is terminated at A'. A surface will thus be obtained consisting of n spherical sheets; like the plane Riemann's surface, it is one over which the n-valued function is a uniform function of the position of the variable point. Fig. M — =-00 To face p. 346. Fig. 57. 177.] CONNECTIVITY OF A RIEMANN's SURFACE 347 But also the connectivity of the n-sheeted spherical surface is the same as that of the n